Factorial

Q. The maximum number of permutations of $2n$ letters in which there are only $a^{\prime }s$ and $b^{\prime }s,$ taken all at a time is given by

1. ${}^{2n}{C}_n$
2. $\frac{2}{1}\cdot \frac{6}{2}\cdot \frac{10}{3}\cdot \cdots \frac{4n-6}{n-1}\cdot \frac{4n-2}{n}$
3. $\frac{n+1}{1}\cdot \frac{n+2}{2}\cdot \frac{n+3}{3}\cdot \frac{n+4}{4}\cdot \cdots \frac{2n-1}{n-1}\cdot \frac{2n}{n}$
4. $\frac{2^n[1\cdot 3\cdot 5\cdots (2n-3\left)\right(2n-1\left)\right]}{n!}$

Q. For $25!$, the correct options is/are

1. exponent of $2$ is $22$
2. exponent of $3$ is $10$
3. exponent of $4$ is $7$
4. exponent of $7$ is $3$

Q. The interval(s) which satisfies the solution set of the inequality $4x-2>x-\frac{3-x}{4}>3$ is/are

1. $\frac{5}{11}<x<3$
2. $x>3$
3. $x<3$
4. $x>\frac{5}{11}$

Q. If $p=1\times 1!+2\times 2!+3\times 3!+...+10\times 10!$, then the remainder when $p+2$ is divided by $11!$ is

Q.

The number of ways in which ten candidates $A_1,A_2,A_3......A_{10}$ can be ranked if $A_1$ and $A_2$ are next to each other is

1. 2!.11!

2. 2!.10!

3. 2!.9!

4. 2!.8!

Q. Seven athletes are participating in a race. If there are three prizes $Gold,Silver$ and $Bronze$, then the number of ways in which the prizes can be distributed to the athletes is

1. $35$
2. $105$
3. $210$
4. $420$

Q. The number of permutations of the word $AUROBIND$ in which vowels appear in the alphabetical order, is

1. ${}^8{P}_4$
2. ${}^8{C}_4$
3. $4!\times {}^8{C}_4$
4. $5!\times {}^8{C}_5$

Q. The value of 1+ 1.1! + 2.2! + 3.3! + .......... + n.n! is

1. (n+1)!+1
2. (n+1)!–1
3. (n+1) !
4. (n–1) !

Q. Total number of different signals that can be made using atleast $3$ flags from $5$ flags of different colours is