## Permutations and Combinations Questions and Answers Part-4

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 25 . 37 . 53 . 72 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 10. The largest three digit prime number dividing $^{2000}C_{1000}$  is  