Untitled Document

Question Bank No: 1

1. Number of positive integral solutions of $x_{1} . x_{2} . x_{3}$ = 30 is

a)		25
b)		26
c)		27
d)		28

2. The number of ways in which 7 persons can sit around a table so that all shall not have the same neighbours in any two arrangements is

a)		180
b)		270
c)		360
d)		720

3. The number of ways in which 6 red roses and 3 white roses can form a garland so that all the white roses come together is

a)		2170
b)		2165
c)		2160
d)		2155

4. Every body in a room shakes hand with every body else. The total number of hand shakes is 66. The total number of persons in the room is

a)		11
b)		12
c)		13
d)		14

5. The number of ways of selecting 10 players out of 22 when 4 of them being excluded and 6 always included is

a)		${}^{22}{C_{10}}$
b)		${}^{12}{C_{2}}$
c)		495
d)		None of these

6. How many different arrangements can be made out of the letters in the expansion $A^{2} B^{3} C^{4}$ , when written in full?

a)		$\frac{9!}{2! 3! 4!}$
b)		2 ! + 3 ! + 4 ! (2 ! + 3 ! + 4 !)
c)		2 ! 3 ! - 4
d)		$\frac{9!}{2! + 3! + 4!}$

7. The number of arrangement which can be made using all the letters of the word LAUGH if the vowels are adjacent is

a)		10
b)		24
c)		120
d)		48

8. In an examination, there are three multiple choice questions and each question has 4 choices. Number of sequences in which a student can fall to get all answers correct is

a)		11
b)		15
c)		80
d)		63

9. If $C_{r} = {}^{n}{C_{r}}$ and S = $\frac{C_{1}}{C_{0}} + 2 \frac{C_{2}}{C_{1}} + 3 \frac{C_{3}}{C_{2}} + .... + n \frac{C_{n}}{C_{n - 1}},$ then

a)		n/S
b)		S is an even integer
c)		n(n+1)/S
d)		None of these

10. The ten's digit of 1 ! + 2 ! + 3! + .... + 49 ! is

a)		1
b)		2
c)		3
d)		4

11. If n = ${}^{m}{C_{2}}$ , then the value of ${}^{n}{C_{2}}$ is given by

a)		${}^{m + 1}{C_{4}}$
b)		${}^{m - 1}{C_{4}}$
c)		${}^{m + 2}{C_{4}}$
d)		None of these

12. If ${}^{56}{P_{r + 6} : {}^{54}{P_{r + 3}}}$ = 30800 : 1, then the value of r is

a)		40
b)		41
c)		42
d)		None of these

13. The smallest value of r satisfying the inequality ${}^{10}{C_{r - 1} > 2. {}^{10}{C_{r}}}$ is

a)		7
b)		10
c)		9
d)		8

14. If 2. ${}^{n}{P_{3}}$ = ${}^{n + 1}{P_{3}}$ , then n is equal to

a)		4
b)		5
c)		6
d)		7

15. The number of rectangles in the adjoining figure is
permutationQ6

a)		5 $\times$ 5
b)		${}^{5}{P_{2} \times {}^{5}{P_{2}}}$
c)		${}^{5}{C_{2} \times {}^{5}{C_{2}}}$
d)		None of these

16. The exponent of 3 in (100) ! is

a)		33
b)		44
c)		48
d)		52

17. If ${}^{n + 1}{C_{5} - {}^{n}{C_{4}}}$ is equal to 56, then the value of n is

a)		At most 7
b)		at least 10
c)		8
d)		None of these

18. The value of ${}^{15}{C_{11}}$ $+$ ${}^{15}{C_{10}}$ is equal to

a)		$\frac{15}{11}$
b)		$\frac{15}{10}$
c)		$\frac{5}{11}$
d)		$\frac{5}{10}$

19. The number if diagonals in a n sided polygon is equal to

a)		${}^{n}{C_{2}}$
b)		${}^{n}{C_{2}}$ – 2
c)		${}^{n}{C_{2}}$ – n
d)		${}^{n}{C_{2}}$ – 1

20. The total number of combinations of n different things taken 1, 2, 3, ......., n at a time is

a)		$2^{n + 1}$
b)		$2^{2 n + 1}$
c)		$2^{n - 1}$
d)		$2^{n} -$ 1

21. The set $S = {1, 2, 3, ...., 12}$ is to be partitioned into three subsets A, B, C of equal size. Thus $A \cup B \cup C = S, A \cap B = B \cap C = C \cap A = φ .$ The number of ways to partition S is

a)		$\frac{12!}{3! {(3!)}^{4}}$
b)		$\frac{12!}{{(4!)}^{3}}$
c)		$\frac{12!}{{(3!)}^{4}}$
d)		$\frac{12!}{3! {(4!)}^{3}}$

22. The letters of the word COCHIN are permuted and all the permutations are arranged in alphabetical order as in English dictionary. The number of words that appear before the word COCHIN is

a)		360
b)		192
c)		96
d)		48

23. At an election, a voter may vote for any number of candidates. not greater than the number to be elected. There are 10 candidates and 4 are to be elected. If a voter votes for atleast one candidate, then the number of ways in which he can vote, is

a)		385
b)		1110
c)		5040
d)		6210

24. If the letters of the word SACHIN are arranged in all possible ways and these words are written in dictionary order, then the word SACHIN appears at serial number

a)		600
b)		601
c)		602
d)		603

25. The number of ways of distributing 8 identical balls in 3 distinct boxes so that no box is empty is

a)		5
b)		$(\begin{matrix} 8 \\ 3 \end{matrix})$
c)		$3^{8}$
d)		21

26. The range of the function $f (x) = {}^{7 - x}{P_{x - 3}}$ is

a)		${1, 2, 3}$
b)		${1, 2, 3, 4}$
c)		${1, 2, 3, 4, 5}$
d)		${1, 2, 3, 4, 5, 6}$

27. A student is to answer 10 out of 13 questions in an examination such that he must choose atleast 4 from the first five questions. The number of choices available to him is

a)		140
b)		196
c)		280
d)		346

28. The number of ways is which 6 men and 5 women can dine at a round table if no two women are to sit together is

a)		6! 5!
b)		30
c)		5! 4!
d)		5! 7!

29. The number of 4 digit odd numbers that can be formed using the digit 0, 1,2, 3, 5, 7 is

a)		216
b)		375
c)		400
d)		720

30. In a class of 100 students there are 70 boys whose average mark in a subject is 75. If the average mark of the complete class is 72, then the average marks of the girls is

a)		73
b)		65
c)		68
d)		74

31. Five digit numbers divisible by 3 are formed using the digits 0, 1, 2, 3, 4, 6, 7 without repetition. The number of such numbers is

a)		936
b)		480
c)		600
d)		216

32. A rectangle with sides $2 m - 1, 2 n - 1$ is divided into squares of unit length by drawing parallel lines as shown in the diagram.
per43
The number of rectangles with odd side length is

a)		${(m + n + 1)}^{2}$
b)		mn (m+1) (n+1)
c)		$m^{2} n^{2}$
d)		$4^{m + n - 1}$

33. When n is a positive integer, then $(n^{2})!$ is

a)		divisible by $n^{2},$ but not by n!
b)		divisible by n!, but not by ${(n!)}^{2}$
c)		divisible by ${(n!)}^{2}$ , but not by ${(n!)}^{n}$
d)		divisible by ${(n!)}^{n}$

34. The product of n natural numbers, n $\geq 2,$ is

a)		not divisible by n
b)		divisible by n, but not by 2n
c)		divisible by 2n, but not by n!
d)		divisible by n!

35. Let $T_{n}$ denote the number of traingles which can be formed using the verticles of a regular polygon of n sides. If $T_{n + 1} - T_{n} = 21,$ then n=

a)		4
b)		5
c)		6
d)		7

36. Let A be a set with n elements. The nunmberof onto functions from A to A, is

a)		$n^{n}$
b)		$n^{n} - n!$
c)		$\frac{n^{n}}{n!}$
d)		n!

37. Let $E = {1, 2, 3, 4}$ and $F = {1, 2}$ . Then the number of onto functions from E to F is

a)		8
b)		12
c)		14
d)		16

38. The number of arrangements of the letters of the word BANANA, in which the two $N^{s} do not appear adjacently$ , is

a)		20
b)		40
c)		60
d)		80

39. The number of ways of arranging six+signs and four - signs in a row, so that no two - signs occur together , is

a)		24
b)		35
c)		44
d)		18

40. The number of 9- digit number formed using the digits 223355888, such that odd digits occupy even places, is

a)		16
b)		36
c)		60
d)		80

41. The number of divisors of the form $4 n + 2, n \geq 0$ of the integer 240 is

a)		3
b)		4
c)		8
d)		10

42. There are 300 students in a college. Every student reads daily 5 newspapers and every newspaper is read by 60 students. The number of newspapers is

a)		atleast 30
b)		atmost 20
c)		exactly 25
d)		none of these

43. Nine hundred distinct n- digit numbers are to be formed using only the 3 digits 2, 5, 7. The smallest value of n for which this is possible is

a)		6
b)		7
c)		8
d)		9

44. Nine boys and 3 girls are to be seated in 2 vans, each having numbered seats, 3 in the front and 4 at the back. The number of ways of seating arrangements, if the girls should sit together in a back row on adjacent seats, is

a)		12!
b)		3 $\cdot 11!$
c)		4 $\cdot 11!$
d)		3!9!

45. Let n and k be positive integers such that $n \geq (\begin{matrix} k + 1 \\ 2 \end{matrix})$ . The number of solutions $(x_{1}, x_{2} ....,, x_{k})$ , $x_{1} \geq 1, x_{2} \geq 2, ....., x_{k} \geq k$ all integers satisfying $x_{1}$ + $x_{2} + ..... x_{k 1} = n$ is

a)		$(\begin{matrix} n - (\begin{matrix} k \\ 2 \end{matrix}) \\ k \end{matrix})$
b)		$(\begin{matrix} n - 1 - (\begin{matrix} k \\ 2 \end{matrix}) \\ k \end{matrix})$
c)		$(\begin{matrix} n - 1 - (\begin{matrix} k \\ 2 \end{matrix}) \\ k - 1 \end{matrix})$
d)		$(\begin{matrix} n + 1 - (\begin{matrix} k \\ 2 \end{matrix}) \\ k - 1 \end{matrix})$

46. There are 4 balls of different colours and 4 boxes of colours same as those of the balls. The number of ways in which the balls, one in each box could be placed such that a ball does not go to a box of its own colour, is

a)		5
b)		6
c)		9
d)		12

47. Eighteen guests have to be seated half on each side of a long table. Four particular guests desire to sit on one particular side and three others on the other side. The number of sitting arrangement is

a)		$(\begin{matrix} 11 \\ 7 \end{matrix}) \cdot {(9!)}^{2}$
b)		$(\begin{matrix} 11 \\ 5 \end{matrix}) \cdot {(9!)}^{2}$
c)		7 $\cdot {(9!)}^{2}$
d)		10!9!

48. A five digit number divisible by 3 is to be formed using the digits 0, 1,2, 3, 4, without rpetition. The number of ways this can be done is

a)		216
b)		184
c)		256
d)		225

49. A student is allowed to select atmost n books from a collection of (2n+1) books. If the number of ways in which he can select atleast one book is 63, then n=

a)		3
b)		4
c)		6
d)		5

50. A box contains 2 white balls, 3 black balls and 4 red balls. The number of ways of drawing 3 balls from the box, if atleast one black ball is included, is

a)		36
b)		42
c)		56
d)		64