Ten different letters of an alphabet are given. Words with five letters are formed from these given letters. Then the number of words which have at least one letter repeated is

a) 30240

b) 69760

c) 99748

d) None of these

Twelve students compete for a race. The number of ways in which first three places can be taken is

a) 12! − 3

b)

c) 12 × 11 × 10

d) 3!

Mary typed a six digit number, but the two 1’s she typed didn’t show. What appeared was 2009. Find the numbers of different six digit numbers she would have typed.

If n is even and
ⁿC₀ < ⁿC₁ < ⁿC₂ < .  .   .< ⁿC_r  > ⁿC_{r + 1} > ⁿC_{r + 2} > .  .  . > ⁿC_n ,

then r is equal to

a)

b)

c)

d)

The number of numbers lying between 40000 and 70000 that can be made from the digits 0, 1, 2, 4, 5, 7 if digits can be repeated in the same number, is

a) 2591

b) 932

c) 766

d) None of these

In a network of railways a small island has fifteen stations. The number of different types of tickets to be printed for each class, if every station must have tickets for other station, is

a) 230

b) 210

c) 340

d) None of these

In a certain test  students gave wrong answers to at least i questions where i = 1, 2, 3, 4, .  .  . k. No student gave more then k wrong answers. The total number of wrong answers given is

a)  +  + ... +  

b)  +  + ... +  

c)  +  + ... +  

d) None of these

In a class tournament where the participants were to play one game with another, two class players fell ill, having played three games each. If the total number of games played is 84, the number of participants at the beginning was

a) 22

b) 15

c) 17

d) None of these

If a denotes the number of permutations of x + 2 things taken all at a time, b the number of permutations of x things taken 11 at a time and c the number of permutations of x – 11 things taken all at a time such that a = 182bc, then the value of x is

a) 15

b) 12

c) 10

d) 19

The sum of all the numbers that can be formed with the digits 2, 3, 4, 5 taken all at a time is equal to

a) 93324

b) 93328

c) 92324

d) None of these

The number of permutations of n different objects taken r (≥3) at a time which include 3 particular objects is

a) ⁿP_r × 3

b) ⁿP_{r − 3} × 3!

c) ^{n − 3}P_{r − 3} × 3!

d) ^rP₃ × ^{n − 3}P_{r – 3}

A library has ‘a’ copies of one book, ‘b’ copies of two books and ‘c’ copies of each of three books and single copy of ‘d’ books. The total number of ways in which these books can be distributed is

a)

b)

c)

d)

How many different arrangements can be made out of the letters in the expansion A²B³C⁴, when written in full?

a)

b)

c) 2! + 3! + 4! (2! 3! 4!)

d) 2! 3! – 4

The number of ways in which n things of which r are alike can be arranged in a circular order is

a)

b)

c)

d)

The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines is

a) 6

b) 18

c) 12

d) 9

The number of diagonals that can be drawn by joining the vertices of an octagon is

a) 28

b) 48

c) 20

d) None of these

In a football championship 153 matches were played. Every two teams played one match with each other. The numbers of teams participating in the championship are

a) 14

b) 22

c) 18

d) None of these

There are n concurrent lines and another line parallel to one of them. The number of different triangles that will be formed by the (n + 1) lines is

a)

b)

c)

d)

The number of diagonals in a polygon of n sides is

a)

b)

c)

d) None of these

In a plane there are two sets of parallel lines, one of m lines and the other of n lines. If the lines of one set cut those of the other, then the number of different parallelograms that can be formed is

a)

b)

c)

d) None of these

A parallelogram is cut by two sets of m lines parallel to its sides. The number of parallelograms then formed is

a)

b)

c)

d) None of these

The straight lines I₁, I₂, I₃ are parallel and lie in the same plane. A total number of m points are taken on I₁, n points on I₂, k points on I₃. The maximum number of triangles formed with vertices at these points is

a) ^{m + n + k}C₃

b) ^{m + n + k}C₃ – ^mC₃ – ⁿC₃ – ^kC₃

c) ^mC₃ + ⁿC₃ + ^kC₃

d) None of these

There are p copies of n different books. The number of different ways in which a non–empty selection can be made from them is

a) (p + 1)ⁿ – 1

b) pⁿ – 1

c) (p + 1)^{n – 1} – 1

d) None of these