Permutation: n Different Things Taken r at a Time

Q.

The number of permutations of the letters of the word $\text{CONSEQUENCE}$ in which all the three $\text{Es}$ arc together, is:

1. $9!3!$

2. $\frac{9!}{2!2!}$

3. $\frac{9!}{2!2!3!}$

4. $\frac{9!}{2!3!}$

Q.

H.C.F and L.C.M. of two numbers are $17$ and $102$ respectively. If one of the numbers is $34$ find the other.

Q. Two numbers are selected simultaneously from the set {6, 7, 8, 9, ………. 39}. If the sum of selected numbers is even then the probability that both the selected numbers are odd, is equal to:

1. $\frac{40}{91}$
2. $\frac{51}{91}$
3. $\frac{8}{33}$
4. $\frac{1}{2}$

Q. If ${}^9{P}_5+5{}^9{P}_4={}^{10}{P}_r,$ then value of $r$ is

1. $3$
2. $4$
3. $5$
4. $6$

Q.

A five digit number divisible by 3 has 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

1. 216

2. 240

3. 600

4. 3125

Q. On the set $Z$ of all integer numbers, define the relation $R$ by $aRb$ iff $a+2b$ is divided by $3$. Then the relation $R$ is

1. only reflexive
2. reflexive and transitive
3. equivalence relation
4. reflexive and symmetric

Q. The number of ways in which $3$ scholarships of unequal value be awarded to $17$ candidates, such that no candidate gets more than one scholarship is

1. ${}^{17}{C}_3$
2. ${17}^3$
3. $3^{17}$
4. ${}^{17}{P}_3$

Q. The number of four digit even numbers that can be formed with 0, 1, 2, 3, 7, 8, is

1. 180
2. 175
3. 160
4. 156

Q. The number of permutations of n dissimilar things taken not more than ‘r’ at a time, when each thing may occur any number of times is

1. $\frac{n(n^r-1)}{n-1)}$
2. $\frac{n(n^n-n^r)}{n-1)}$
3. ${}^n{P}_1{+}^n{P}_2+\cdots \cdots {+}^n{P}_r$
4. $\frac{n{(n-1)}^r}{n-1}$

Q. A person draws a card from a pack, replaces it shuffles the pack, again draws a card, replace it and draws again. This process he does until he draw a heart card. The probability that he will have to make at least four draws is

1. $\frac{27}{256}$
2. $\frac{27}{16}$
3. $\frac{17}{256}$
4. $\frac{27}{64}$