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Beyond Advanced · Permutations and Combinations

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1
Let Z(n)Z(n) denote the number of trailing zeros of n!n! (for example Z(24)=4Z(24)=4 and Z(25)=6Z(25)=6). As nn runs over 1,2,3,…,10001,2,3,\ldots,1000, how many different values does Z(n)Z(n) take?
Not attempted
Your answer: —
Correct answer: 201
2
Let Sn=1⋅1!+2⋅2!+3⋅3!+⋯+n⋅n!S_n=1\cdot 1!+2\cdot 2!+3\cdot 3!+\cdots+n\cdot n!. Find the smallest positive integer nn for which Sn+1S_n+1 is divisible by 10710^{7}.
Not attempted
Your answer: —
Correct answer: 29
3
How many 55-digit numbers can be formed using the digits 0,1,2,3,4,50,1,2,3,4,5 with no digit repeated, such that the number is divisible by 44?
Not attempted
A
168168
B
126126
C
144144
D
312312
4
Which of the following statements about factorials and binomial coefficients are true?
Not attempted
A
The decimal expansion of (10050)\binom{100}{50} does not end in the digit 00.
B
The highest power of 1212 that divides 50!50! is 122312^{23}.
C
The product 1⋅3⋅5⋯97⋅991\cdot3\cdot5\cdots 97\cdot 99 of all odd numbers up to 9999 is divisible by 5135^{13}.
D
The highest power of 22 that divides 1024!1024! is 210242^{1024}.
5
Find the number of ordered pairs (a,b)(a,b) of positive integers satisfying a!⋅b!=10!a!\cdot b!=10!.
Not attempted
A
44
B
22
C
33
D
66
6
Which of the following statements about divisibility of factorials are true?
Not attempted
A
For every composite integer n≥6n\ge6, n2n^2 divides n!n!.
B
Exactly 3434 integers nn with 1≤n≤501\le n\le50 satisfy n2∣n!n^2\mid n!.
C
For every prime p≥3p\ge3, p2p^2 divides (2p)!(2p)! but p3p^3 does not.
D
For every positive integer nn, (n!)n+1(n!)^{n+1} divides (n2)!(n^2)!.
7
How many integers from 10001000 to 99999999 inclusive are divisible by 55 and have at least two equal digits?
Not attempted
A
792792
B
952952
C
12961296
D
848848
8
Consider all distinct arrangements of the 1111 letters of the word ARRANGEMENT\text{ARRANGEMENT}. In how many of them does every letter AA stand to the left of every letter EE?
Not attempted
Your answer: —
Correct answer: 415800
9
A row a1,a2,…,a8a_1,a_2,\ldots,a_8 is a rearrangement of 1,2,…,81,2,\ldots,8 that first strictly increases up to the entry 88 and then strictly decreases (either part may be empty). Let U\mathcal U be the set of all such rows. Which statements are true?
Not attempted
A
∣U∣=128|\mathcal U|=128.
B
Exactly 2121 rows of U\mathcal U have 88 in the third position.
C
Exactly 1616 rows of U\mathcal U begin with 44.
D
In every row of U\mathcal U the entry 11 is at one of the two ends.
10
Eight distinct people, among them AA, BB and CC, stand in a row. Which of the following counts of arrangements are correct?
Not attempted
A
AA, BB, CC stand in this left-to-right order (not necessarily adjacent): 67206720 arrangements.
B
BB stands somewhere between AA and CC: 1344013440 arrangements.
C
No two of AA, BB, CC stand next to each other: 1440014400 arrangements.
D
AA and BB stand next to each other, but CC stands next to neither of them: 86408640 arrangements.
11
Find the sum of all 55-digit numbers that can be formed using each of the digits 0,1,2,3,40,1,2,3,4 exactly once.
Not attempted
A
26666402666640
B
25999802599980
C
21333122133312
D
20000402000040
12
Five-letter words (meaningful or not) are formed using the letters of BALLOON\text{BALLOON}, each letter being used at most as many times as it occurs there. Which statements are true?
Not attempted
A
The number of different such words is 690690.
B
Exactly 360360 of these words contain both the letter BB and the letter NN.
C
Exactly 540540 of these words contain the letter BB.
D
Exactly 144144 of these words contain the two letters LL next to each other.
13
A palindromic arrangement of all 1111 letters of MISSISSIPPI\text{MISSISSIPPI} is one that reads the same from left to right and from right to left. How many such arrangements have no two letters SS next to each other?
Not attempted
A
1818
B
3030
C
1212
D
3636
14
Twelve chairs numbered 1,2,…,121,2,\ldots,12 stand in a row, as shown (chairs 55 and 66 are broken and cannot be used; the others are intact). Four different people choose four of the intact chairs, no two of them on chairs whose numbers differ by 11. In how many ways can the four people be seated?
Not attempted
123456789101112Chairs 5 and 6 are broken
A
840840
B
30243024
C
4646
D
11041104
15
All 88 letters of the word EQUATION\text{EQUATION} (all distinct) are arranged in a row. Which statements are true?
Not attempted
A
The five vowels occupy five consecutive places in exactly 28802880 arrangements.
B
No arrangement has no two vowels adjacent.
C
The vowels appear in alphabetical order A,E,I,O,UA,E,I,O,U in exactly 336336 arrangements.
D
The vowels occupy exactly the odd-numbered places in exactly 720720 arrangements.
16
The letters of BANANAS\text{BANANAS} (BB, A3A^3, N2N^2, SS; 420420 distinct arrangements in all) are arranged in a row. List-I gives a condition on the arrangement; List-II gives the number of arrangements satisfying it. Match each condition with its count. List-IList-IIP) A’s together1) 36Q) no two A’s adjacent2) 60R) A’s together, N’s apart3) 120S) B before S4) 1805) 210\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)}\ \text{A's together} & \text{1)}\ 36 \\ \text{Q)}\ \text{no two A's adjacent} & \text{2)}\ 60 \\ \text{R)}\ \text{A's together, N's apart} & \text{3)}\ 120 \\ \text{S)}\ \text{B before S} & \text{4)}\ 180 \\ & \text{5)}\ 210 \end{array}
Not attempted
A
P-2, Q-4, R-1, S-3
B
P-2, Q-3, R-1, S-5
C
P-1, Q-3, R-2, S-5
D
P-5, Q-3, R-4, S-1
17
Four married couples are to sit at a round table with 88 identical seats so that men and women alternate and no husband sits next to his wife. Two seatings that differ by a rotation are regarded as the same. In how many ways can this be done?
Not attempted
MWMWMWMW
A
144144
B
2424
C
1212
D
7272
18
A bracelet is made by threading 77 beads on a circular string: 33 identical red beads, 22 identical blue beads and 22 identical white beads. Two bracelets are the same if one can be obtained from the other by a rotation or by turning the bracelet over. How many different bracelets are possible?
Not attempted
Your answer: —
Correct answer: 18
19
Eight distinct people A,B,C,…A,B,C,\ldots sit at a round table with 88 identical seats; rotations of a seating are regarded as the same, but a seating and its mirror image are different. Which statements are true?
Not attempted
A
AA and BB are not neighbours in exactly 36003600 seatings.
B
BB has both AA and CC as neighbours in exactly 240240 seatings.
C
AA and BB sit directly opposite each other in exactly 14401440 seatings.
D
No two of AA, BB, CC are neighbours in exactly 14401440 seatings.
20
Eight different people, including AA, BB and CC, sit round the circular table shown, which has 88 identical equally spaced seats (opposite seats are joined by the dashed lines). Rotations of a seating are regarded as the same, and mirror images are different. In how many seatings are AA, BB and CC such that no two of them are neighbours and no two of them sit directly opposite each other?
Not attempted
Dashed lines join opposite seats
A
14401440
B
57605760
C
360360
D
720720
21
A jeweller has 99 beads, all of different colours, including a gold bead AA and a silver bead BB. A bracelet is made by threading 66 of the beads on a circular string so that AA and BB are both used and are next to each other. Two bracelets are the same if one is obtained from the other by rotating or turning it over. How many different bracelets can be made?
Not attempted
AB
Your answer: —
Correct answer: 840
22
Ten distinct people, including AA, BB, CC and DD, sit at a round table with 1010 identical seats; rotations are regarded as the same and mirror images are different. In how many seatings do exactly three people sit between AA and BB along the shorter of the two arcs joining them, with CC among those three people and DD sitting next to AA?
Not attempted
A
Your answer: —
Correct answer: 7200
23
Beads of different colours are strung on a circular string; two necklaces are the same if one can be rotated or turned over to give the other. Which statements are true?
Not attempted
A
With 66 beads of different colours, 6060 different necklaces can be made.
B
With 55 beads of different colours, exactly 1212 necklaces have two particular beads next to each other.
C
Choosing 44 beads out of 77 different ones, there are 210210 necklaces.
D
With 33 identical red beads and 33 beads of three other different colours, there are 2020 necklaces.
24
A committee of 66 is chosen from 77 men M1,…,M7M_1,\ldots,M_7 and 55 women W1,…,W5W_1,\ldots,W_5. It must contain at least 22 and at most 44 women. Also M1M_1 and W1W_1 refuse to serve together, and W2W_2 will serve only if M2M_2 also serves. How many committees are possible?
Not attempted
A
611611
B
426426
C
565565
D
427427
25
Evaluate ∑k=06(6k)(9k)\displaystyle\sum_{k=0}^{6}\binom{6}{k}\binom{9}{k}.
Not attempted
Your answer: —
Correct answer: 5005
26
Twenty points are marked on a circle at equal spacing. In how many ways can 44 of them be chosen so that no two chosen points are neighbours on the circle?
Not attempted
Your answer: —
Correct answer: 2275
27
How many 44-element subsets of {1,2,…,12}\{1,2,\ldots,12\} do not contain three consecutive integers (that is, no xx with x,x+1,x+2x,x+1,x+2 all in the subset)?
Not attempted
A
405405
B
495495
C
414414
D
396396
28
A hand of 55 cards is dealt from a standard pack of 5252 cards (1313 cards in each of 44 suits, with exactly one ace per suit). How many different hands contain at least one card of every suit and at least one ace?
Not attempted
A
229272229272
B
456192456192
C
685464685464
D
263640263640
29
Three consecutive entries (nr−1),(nr),(nr+1)\binom{n}{r-1},\binom{n}{r},\binom{n}{r+1} (1≤r≤n−11\le r\le n-1) in the row nn of Pascal's triangle can form an arithmetic progression for some rr only for certain nn. How many integers nn with 1≤n≤1001\le n\le100 have this property?
Not attempted
Your answer: —
Correct answer: 8
30
Which of the following evaluations of sums of binomial coefficients are correct?
Not attempted
A
∑k=010k(10k)=5120\displaystyle\sum_{k=0}^{10}k\binom{10}{k}=5120.
B
∑k=010k2(10k)=25600\displaystyle\sum_{k=0}^{10}k^2\binom{10}{k}=25600.
C
∑k=010(10k)2=(209)\displaystyle\sum_{k=0}^{10}\binom{10}{k}^2=\binom{20}{9}.
D
∑k=010(−1)kk2(10k)=0\displaystyle\sum_{k=0}^{10}(-1)^k k^2\binom{10}{k}=0.
31
How many 44-element subsets of {1,2,…,12}\{1,2,\ldots,12\} have a sum divisible by 33?
Not attempted
Your answer: —
Correct answer: 165
32
How many ordered pairs (a,b)(a,b) of positive integers are there with lcm⁡(a,b)=24⋅33⋅52⋅7\operatorname{lcm}(a,b)=2^4\cdot3^3\cdot5^2\cdot7 and gcd⁡(a,b)\gcd(a,b) divisible by 66?
Not attempted
A
945945
B
525525
C
735735
D
263263
33
Which statements about divisors and selections are true?
Not attempted
A
The sum of all positive divisors of 72007200 is 2538925389.
B
72007200 can be written as a product abab of two coprime positive integers with a≤ba\le b in exactly 88 ways.
C
The product of all positive divisors of 72007200 equals 7200547200^{54}.
D
A shop has 66 identical pens, 44 identical pencils and 33 different erasers. The number of ways of choosing at least one item is 280280.
34
Let N=210⋅36⋅54⋅73N=2^{10}\cdot3^{6}\cdot5^{4}\cdot7^{3}. Which of the following statements are true?
Not attempted
A
NN has exactly 144144 positive divisors that are perfect squares.
B
NN has exactly 4848 positive divisors that are perfect cubes.
C
NN has exactly 140140 positive divisors that are perfect squares but not perfect cubes.
D
NN has exactly 13521352 positive divisors that are neither perfect squares nor perfect cubes.
35
For a positive integer NN, τ(N)\tau(N) denotes the number of its positive divisors. List-I gives a value of τ(N)\tau(N); List-II gives candidate values of NN. Match each item of List-I with the smallest positive integer NN having that number of divisors. List-IList-IIP) τ(N)=121) 48Q) τ(N)=102) 60R) τ(N)=153) 120S) τ(N)=164) 1445) 192\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)}\ \tau(N)=12 & \text{1)}\ 48 \\ \text{Q)}\ \tau(N)=10 & \text{2)}\ 60 \\ \text{R)}\ \tau(N)=15 & \text{3)}\ 120 \\ \text{S)}\ \tau(N)=16 & \text{4)}\ 144 \\ & \text{5)}\ 192 \end{array}
Not attempted
A
P-5, Q-1, R-4, S-2
B
P-2, Q-3, R-5, S-1
C
P-2, Q-1, R-4, S-3
D
P-3, Q-1, R-2, S-4
36
How many positive integers n≤200n\le200 have exactly 1212 positive divisors?
Not attempted
Your answer: —
Correct answer: 14
37
A bag contains 55 identical red balls, 44 identical blue balls, 33 identical green balls and 33 balls of three different shades of black. How many different selections of exactly 88 balls are there that contain at least one ball of each of the four colours (red, blue, green, black)?
Not attempted
Your answer: —
Correct answer: 66
38
Find the sum of all positive divisors of 10!10! that are divisible by 44 but not divisible by 88.
Not attempted
Your answer: —
Correct answer: 120032
39
How many triples (x,y,z)(x,y,z) of integers with 0≤x<y<z0\le x<y<z satisfy x+y+z=30x+y+z=30?
Not attempted
Your answer: —
Correct answer: 75
40
Find the number of ordered quadruples (x,y,z,w)(x,y,z,w) of non-negative even integers, each at most 1010, with x+y+z+w=24x+y+z+w=24.
Not attempted
A
125125
B
455455
C
119119
D
29252925
41
Ordinary six-faced dice are rolled; the dice are distinguishable (for example, of different colours). Which statements are true?
Not attempted
A
With four dice, there are 146146 outcomes whose faces sum to 1414.
B
With three dice, there are 2727 outcomes whose faces sum to 1010.
C
With three dice, there are 66 outcomes whose faces sum to 1010 if outcomes that differ only in which die shows which number are regarded as the same.
D
With four dice, there are 7070 outcomes whose faces sum to 1414 with every die showing at least 22.
42
Each option shows five boxes (the circles) and some segments joining pairs of boxes. Nine identical balls are placed in the five boxes (a box may contain any number of balls, including none) so that no two boxes joined by a segment are both empty. For which diagram is the number of different ways to place the balls the greatest?
Not attempted
A
five boxes in a chain\text{five boxes in a chain}
B
five boxes in a pentagon\text{five boxes in a pentagon}
C
two triangles sharing one box\text{two triangles sharing one box}
D
one box joined to each of the other four\text{one box joined to each of the other four}
43
Find the number of ordered triples (a,b,c)(a,b,c) of positive integers such that abc=27⋅34⋅53abc=2^{7}\cdot3^{4}\cdot5^{3} and each of a,b,ca,b,c is even.
Not attempted
Your answer: —
Correct answer: 2250
44
Six balls are placed into three boxes. List-I describes the balls, the boxes and the rule on box contents; List-II gives the number of different distributions. Match each item with its count. List-IList-IIP) 6 identical, 3 distinct boxes, each box≤31) 3Q) 6 identical, 3 identical boxes, each box≤32) 10R) 6 distinct, 3 distinct boxes, none empty3) 90S) 6 distinct, 3 identical boxes, none empty4) 5405) 15\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)}\ \text{6 identical, 3 distinct boxes, each box}\le3 & \text{1)}\ 3 \\ \text{Q)}\ \text{6 identical, 3 identical boxes, each box}\le3 & \text{2)}\ 10 \\ \text{R)}\ \text{6 distinct, 3 distinct boxes, none empty} & \text{3)}\ 90 \\ \text{S)}\ \text{6 distinct, 3 identical boxes, none empty} & \text{4)}\ 540 \\ & \text{5)}\ 15 \end{array}
Not attempted
A
P-2, Q-1, R-3, S-4
B
P-5, Q-2, R-4, S-3
C
P-2, Q-1, R-4, S-3
D
P-1, Q-3, R-2, S-4
45
Let SS be the set of ordered quadruples (x,y,z,w)(x,y,z,w) of non-negative integers with x+y+z+w=20x+y+z+w=20. Which statements are true?
Not attempted
A
∣S∣=1771|S|=1771.
B
Exactly 946946 members of SS satisfy x≥yx\ge y.
C
Exactly 825825 members of SS satisfy x<yx<y.
D
Exactly 108108 members of SS satisfy x≥y≥z≥wx\ge y\ge z\ge w.
46
The expression (a+b+c+d)10(a+b+c+d)^{10} is expanded and like terms are collected. A monomial is a term apbqcrdsa^{p}b^{q}c^{r}d^{s} with p+q+r+s=10p+q+r+s=10. Which statements are true?
Not attempted
A
The expansion has 286286 different monomials.
B
Exactly 206206 of the monomials have every exponent at most 66.
C
Exactly 120120 of the monomials contain all four letters (every exponent at least 11).
D
Exactly 336336 of the monomials have exactly one exponent equal to 00.
47
In how many ways can 1212 different books be divided into four groups, two of size 44 and two of size 22, if the groups are not labelled (only the contents of the groups matter)?
Not attempted
A
5197551975
B
207900207900
C
103950103950
D
3465034650
48
Nine different gifts are distributed among three children (the children are distinguishable) so that every child receives at least 22 gifts. In how many ways can this be done?
Not attempted
A
1815018150
B
1123511235
C
92409240
D
1150811508
49
Ten points P1,P2,…,P10P_1,P_2,\ldots,P_{10} lie on a circle in this order (see figure). The points are to be joined in five pairs by five chords, each point being an end of exactly one chord, so that no two chords cross inside the circle. Moreover the chord P1P6P_1P_6 (dashed in the figure) must not be one of the five chords. How many such sets of five chords are there?
Not attempted
P1P2P3P4P5P6P7P8P9P10
A
4242
B
3838
C
44
D
105105
50
Ten players, among them two pairs of rivals (A,B)(A,B) and (C,D)(C,D), are to be divided into five pairs to play five matches (the order of the matches is irrelevant, as is the order within a pair). In how many ways can the pairing be done so that neither AA and BB nor CC and DD are paired with each other? (In the figure the dashed segments join the two pairs of rivals.)
Not attempted
AECFGBHDIJ
Your answer: —
Correct answer: 750
51
Twelve different students are divided into four groups of three, and the groups are assigned to four different laboratories (one group per laboratory). Each group then chooses one of its three members as its leader. Two particular students AA and BB must work in the same laboratory and neither of them may be a leader. In how many ways can the whole arrangement (groups, laboratories and leaders) be made?
Not attempted
Your answer: —
Correct answer: 1814400
52
Which of the following counts of divisions into groups are correct?
Not attempted
A
Fifteen different people can be divided into three teams of five (the teams not being named) in 126126126126 ways.
B
Ten different books can be given to three named students so that two of them get 44 books each and the third gets 22 books in 94509450 ways.
C
Eight players can be paired into four matches so that two particular players are not paired together in 9090 ways.
D
Six boys and six girls can be divided into six pairs, each consisting of one boy and one girl, in 720720 ways.
53
Eight different books are divided as described in List-I. Unless named boys are mentioned, the groups are not labelled. List-II gives numbers of ways. Match each item of List-I with its count. (In R, the three named boys receive 3,3,23,3,2 books in some order, every order allowed.) List-IList-IIP) two teams of 41) 35Q) groups of 3, 3, 22) 105R) 3, 3, 2 to three named boys3) 280S) four pairs4) 16805) 560\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)}\ \text{two teams of 4} & \text{1)}\ 35 \\ \text{Q)}\ \text{groups of 3, 3, 2} & \text{2)}\ 105 \\ \text{R)}\ \text{3, 3, 2 to three named boys} & \text{3)}\ 280 \\ \text{S)}\ \text{four pairs} & \text{4)}\ 1680 \\ & \text{5)}\ 560 \end{array}
Not attempted
A
P-1, Q-3, R-4, S-2
B
P-1, Q-5, R-3, S-2
C
P-2, Q-3, R-5, S-1
D
P-5, Q-4, R-3, S-1
54
A permutation π\pi of {1,2,…,8}\{1,2,\ldots,8\} has a fixed point ii if π(i)=i\pi(i)=i. How many permutations of {1,2,…,8}\{1,2,\ldots,8\} have at least 55 fixed points?
Not attempted
Your answer: —
Correct answer: 141
55
How many integers nn with 1≤n≤10001\le n\le1000 are divisible by none of 2,3,5,72,3,5,7?
Not attempted
Your answer: —
Correct answer: 228
56
Six teachers T1,…,T6T_1,\ldots,T_6 each wrote one exam paper (paper ii was written by TiT_i). The papers are shuffled and handed back one to each teacher for marking, so that no teacher gets their own paper, and also T1T_1 does not get the paper of T2T_2 and T2T_2 does not get the paper of T1T_1. In how many ways can the papers be handed out? (In the figure, row ii counted from the top is teacher TiT_i and column jj is paper jj; a shaded cell is a forbidden handing-over.)
Not attempted
Your answer: —
Correct answer: 168
57
In how many ways can the eight digits 1,1,2,2,3,3,4,41,1,2,2,3,3,4,4 be arranged in a row so that no two equal digits are next to each other?
Not attempted
A
10801080
B
13441344
C
840840
D
864864
58
How many functions f:{1,2,3,4,5,6}→{a,b,c,d}f:\{1,2,3,4,5,6\}\to\{a,b,c,d\} are onto (every element of the codomain is hit) and satisfy f(1)≠f(2)f(1)\ne f(2)?
Not attempted
A
15601560
B
13201320
C
11701170
D
240240
59
Four married couples sit at a round table with 88 identical seats. Rotations of a seating are regarded as the same (mirror images are different). In how many seatings does no husband sit next to his wife?
Not attempted
A
14401440
B
50405040
C
14881488
D
21602160
60
DnD_n denotes the number of derangements of nn objects (permutations with no fixed point). Which statements are true?
Not attempted
A
D7=1854D_7=1854.
B
For n≥1n\ge1, DnD_n is odd exactly when nn is odd.
C
For even n≥2n\ge2, Dn=nDn−1−1D_n=nD_{n-1}-1.
D
The number of permutations of {1,…,7}\{1,\ldots,7\} with exactly three fixed points is (73)D5=1540\binom73D_5=1540.
61
Match each counting problem of List-I with its answer in List-II. (In S, positions 1,3,51,3,5 must not hold 1,3,51,3,5 respectively; positions 22 and 44 are unrestricted. The figure shows the allowed and forbidden cells for S: row ii from the top is the object ii, column jj is the position, and shaded cells are forbidden.) List-IList-IIP) perms of 6 with no fixed point1) 64Q) perms of 6, exactly 2 fixed2) 135R) onto maps, 5-set to 3-set3) 150S) perms of 5, none fixed at 1, 3, 54) 2655) 44\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)}\ \text{perms of 6 with no fixed point} & \text{1)}\ 64 \\ \text{Q)}\ \text{perms of 6, exactly 2 fixed} & \text{2)}\ 135 \\ \text{R)}\ \text{onto maps, 5-set to 3-set} & \text{3)}\ 150 \\ \text{S)}\ \text{perms of 5, none fixed at 1, 3, 5} & \text{4)}\ 265 \\ & \text{5)}\ 44 \end{array}
Not attempted
A
P-5, Q-2, R-3, S-4
B
P-4, Q-1, R-3, S-2
C
P-1, Q-4, R-2, S-3
D
P-4, Q-2, R-3, S-1
62
Four lattice diagrams are shown in the options. Each shows the points (i,j)(i,j) with 0≤i≤50\le i\le5, 0≤j≤40\le j\le4, joined by grid lines, with S=(0,0)S=(0,0) at the bottom-left and T=(5,4)T=(5,4) at the top-right; two lattice points (black dots) are blocked. A path from SS to TT uses only unit steps to the right or up along grid lines and may not visit a blocked point. In which diagram is the number of such paths the greatest?
Not attempted
A
blocked: (1,3) and (3,3)\text{blocked: }(1,3)\text{ and }(3,3)
ST
B
blocked: (1,2) and (2,3)\text{blocked: }(1,2)\text{ and }(2,3)
ST
C
blocked: (1,2) and (4,1)\text{blocked: }(1,2)\text{ and }(4,1)
ST
D
blocked: (2,1) and (3,2)\text{blocked: }(2,1)\text{ and }(3,2)
ST
63
A path goes from (0,0)(0,0) to (8,4)(8,4) using only unit steps to the right or up. In how many such paths does every lattice point (x,y)(x,y) visited satisfy x≥yx\ge y (the path never goes strictly above the line y=xy=x)?
Not attempted
ST
Your answer: —
Correct answer: 275
64
Each option shows a board made of 6×46\times4 unit cells (6 columns, 4 rows) with one cell removed (shaded). A rectangle is counted if its sides lie along grid lines and it does not cover the removed cell (it may touch it). On which board can the greatest number of rectangles be drawn? (Columns are counted from the left and rows from the bottom.)
Not attempted
A
hole in column 2, row 1\text{hole in column }2\text{, row }1
B
hole in column 3, row 1\text{hole in column }3\text{, row }1
C
hole in column 1, row 2\text{hole in column }1\text{, row }2
D
hole in column 2, row 2\text{hole in column }2\text{, row }2
65
Three vertices of a regular 1515-gon are chosen to form a triangle. How many of the (153)\binom{15}{3} triangles contain the centre of the 1515-gon in their interior?
Not attempted
Your answer: —
Correct answer: 140
66
A convex decagon (ten vertices) has all its diagonals drawn, and no three diagonals meet at a point inside it. Into how many regions do the diagonals divide the interior of the decagon?
Not attempted
A
246246
B
211211
C
245245
D
281281
67
Lines are drawn in a plane; regions are the parts into which the plane is divided. Which statements are true?
Not attempted
A
1212 lines, no two parallel and no three through a common point, divide the plane into 7979 regions.
B
1212 lines, of which exactly 55 are parallel to one another and the remaining 77 are in general position (no other parallels, no three concurrent), divide the plane into 6767 regions.
C
1010 distinct lines through one point divide the plane into 1111 regions.
D
1212 lines, exactly 44 of which pass through one point and the rest in general position, divide the plane into 7676 regions.
68
The figure shows the 2525 lattice points of a 5×55\times5 array, with the central point removed (the open circle is not available). How many triangles (non-degenerate) have all three vertices among the remaining 2424 points?
Not attempted
A
21482148
B
19001900
C
19361936
D
19081908
69
Twelve points lie in a plane: exactly 55 of them lie on one line ℓ1\ell_1, exactly 44 others lie on a second line ℓ2\ell_2, the remaining 33 points are placed so that no other three of the twelve points are collinear. Which of the following statements are true?
Not attempted
line with 5 pointsline with 4 points(not to scale)
A
The twelve points determine exactly 5252 different lines.
B
The twelve points are vertices of exactly 206206 triangles.
C
Four parallel lines are cut by five other parallel lines (not parallel to the first four); the figure formed contains exactly 120120 parallelograms.
D
A convex octagon with all diagonals drawn and no three diagonals concurrent has exactly 5656 interior intersection points of diagonals.
70
Match each item of List-I (points are in a plane; apart from the stated collinear four, no three of the nine points are collinear; the 4×44\times4 array has 1616 lattice points) with its count in List-II. List-IList-IIP) lines through 9 points, 4 collinear1) 20Q) triangles from 9 points, 4 collinear2) 31R) diagonals of a 12-gon3) 54S) squares, any tilt, on a 4×4 array4) 805) 36\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)}\ \text{lines through 9 points, 4 collinear} & \text{1)}\ 20 \\ \text{Q)}\ \text{triangles from 9 points, 4 collinear} & \text{2)}\ 31 \\ \text{R)}\ \text{diagonals of a 12-gon} & \text{3)}\ 54 \\ \text{S)}\ \text{squares, any tilt, on a }4\times4\text{ array} & \text{4)}\ 80 \\ & \text{5)}\ 36 \end{array}
Not attempted
9 points: 4 on one line4 x 4 array of points
A
P-5, Q-4, R-3, S-2
B
P-2, Q-1, R-3, S-4
C
P-2, Q-4, R-3, S-1
D
P-4, Q-2, R-1, S-3
For lines and triangles correct the unrestricted counts for the collinear points; for tilted squares count by the square that bounds them.
P: (92)−(42)+1=36−6+1=31\binom92-\binom42+1=36-6+1=31. Q: (93)−(43)=84−4=80\binom93-\binom43=84-4=80. R: 12⋅92=54\dfrac{12\cdot9}{2}=54. S: a tilted square inscribed in an axis-parallel square of side ss lattice units can be placed with its vertices at k=0,…,s−1k=0,\ldots,s-1 along each side, i.e. ss squares (counting the axis-parallel one); there are (4−s)2(4-s)^2 bounding squares of side ss: ∑s=13s(4−s)2=9+8+3=20\sum_{s=1}^{3}s(4-s)^2=9+8+3=20. Key: P–2, Q–4, R–3, S–1; the value 36=(92)36=\binom92 is a decoy.