1. The number of ways of arranging letters
of the word RACHIT so that the vowels are in alphabetical
order is

a) 120

b) 240

c) 360

d) 480

Explanation: We can arrange letters of the word RACHIT in 6! ways. Out of these exactly half, that is, there are

2. if \[^{n-1}C_{r}=\left(K^{2}-3\right)\left(^{n}C_{r+1}\right)\] then K belong to

a) \[\left[-\sqrt{3},\sqrt{3}\right]\]

b) \[\left(-\infty,-2\right)\]

c) (2, \[\infty\] )

d) \[(\sqrt{3}\] ,2]

Explanation:

3. A five digit number divisible by 3 is to be formed using the numerals 0, 1, 2, 3, 4 and 5 without
repetition. The total number of ways in which this can be
done is

a) 216

b) 240

c) 600

d) 3125

Explanation: The sum of the numerals 0, 1, 2, 3, 4 and 5 is

4. The number of ways in which a mixed
double game can be arranged from amongst 9 married
couples if no husband and wife play in the same game is

a) 756

b) 1512

c) 3024

d) 378

Explanation: We can choose two men out of 9 in \[^{9}C_{2}\] ways

5. The sum of the divisors of \[2^{5}.3^{7}.5^{3}.7^{2}\] is

a) \[2^{6}.3^{8}.5^{4}.7^{3}\]

b) \[2^{6}.3^{8}.5^{4}.7^{3}-2.3.5.7\]

c) \[2^{6}.3^{8}.5^{4}.7^{3}-1\]

d) none of these

Explanation: Any divisor of 2

^{5}. 3

^{7}. 5

^{3}. 7

^{2}is of the form

6. The number of times the digit 3 will be
written when listing the integers from 1 to 1000 is

a) 269

b) 300

c) 271

d) 302

Explanation: Since 3 does not occur in 1000, we have to

7. The number of positive integers n such
that \[2^{n}\] divides n! is

a) exactly 1

b) exactly 2

c) infinite

d) none of these

Explanation: The exponent of 2 in n! is given by

8. The sum\[\sum\sum_{0\leq i< j\leq 10}\left(^{10}C_{j}\right)\left(^{j}C_{i}\right)\] is equal to

a) \[2^{10}-1\]

b) \[2^{10}\]

c) \[3^{10}-1\]

d) \[3^{10}\]

Explanation:

9. A class contains 4 boys and g girls. Every sunday five students , including at least three boys go for
a picnic to Appu Ghar, a different group being sent every
week. During, the picnic, the class teacher gives each girl
in the group a doll. If the total number of dolls distributed
was 85, then value of g is

a) 15

b) 12

c) 8

d) 5

Explanation: Number of groups having 4 boys and 1 girl

10. The largest three digit prime number
dividing \[^{2000}C_{1000}\] is

a) 661

b) 659

c) 673

d) 681

Explanation: