Factorial

Q.

How many words, with or without meaning, can be formed by using all the letters of the word `DELHI', using each letter exactly once ?

Q. The sum of all the numbers of four different digit that can be made by using the digits $0,1,2$ and $3$ is

1. $39996$
2. $38664$
3. none of these
4. $26664$

Q.

The number of ways in which ten candidates $A_1,A_2,...,A_{10}$ can be ranked, if $A_1$ is always above $A_{10}$ is?

1. $5!$

2. $2\left(5!\right)$

3. $10!$

4. $\frac{10!}{2}$

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 number of arrangements of the word "DELHI" in which E precedes 1 is

1. 30

2. 60

3. 120

4. 59

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.

If $\frac{\left(2n\right)!}{3!(2n-3)!}and\frac{n!}{2!(n-2)!}$ are in the ratio 44:3, find n.

Q.

Find the total number of ways of selecting five letters from the letters of the word INDEPENDENT.___

Q.

If (n+2)!= 60 [(n-1)!], find n.

Q.

The product of r consecutive positive integers is divisible by

1. (r-1)!

2. None of these

3. (r+1)!

4. r!

Q.

Prove that: $\frac{1}{9!}+\frac{1}{10!}+\frac{1}{11!}=\frac{122}{11!}$

Q. Evaluate: ${\int }_0^2(e^x+\frac{1}{2x+1})dx$

1. $e^2-1$
2. $e^2+\frac{ln5}{2}$
3. $e^2-1+\frac{ln5}{2}$
4. $e^2-\frac{ln5}{2}+1$

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.

Prove that : n! (n+2) = n! + (n+1)!

Q. $A,B,C$ are three persons among $7$ persons who speak at a function. The number of ways in which it can be done if ${}^{\prime }{A}^{\prime }$ speaks before ${}^{\prime }{B}^{\prime }$ and ${}^{\prime }{B}^{\prime }$ speaks before ${}^{\prime }{C}^{\prime }$ is

Q.

Ten persons , amongst whom are $A$ , $B$ and $C$ to speak at a function. The number of ways in which it can be done if $A$ wants to speak before $B$ and $B$ wants to speak before $C$ is?

1. $\frac{10!}{6}$

2. $3!+7!$

3. ${}^{10}{P}_3.7!$

4. None of these

Q.

Show that $(a-b)\times (a+b)=2(a\times b)$

Q.

There are two works each of 3 volumes and two works each of 2 volumes ; In how many ways can the 10 books be placed on a shelf so that the volumes of the same work are not separated?

Q. The largest power of $2$ that divides $\frac{200!}{100!}$

1. $99$
2. $98$
3. $101$
4. $100$

Q.

If (n+1)!= 90 [(n-1)!], find n.

Q.

The number of ways in which the following prizes will be given to a class of $20$ boys , first and second Mathematics , first and second Physics, first Chemistry and first English is?

1. ${20}^4\times {19}^2$

2. ${20}^3\times {19}^3$

3. ${20}^2\times {19}^4$

4. None of these

Q. If 100! is divided by $(72{)}^k$, where $k\in N),$ then the maximum value of $k$ is

Q. The number of ways in which the time table for Monday can be made if there must be $5$ lessons to be taught that day (Algebra, Geometry, Calculus, Trigonometry, Vector) and topics Algebra and Geometry cannot immediately follow each other are

1. $72$
2. $5!$
3. $3\times \frac{5!}{2}$
4. $6!$

Q.

Consider the following statements. Let $A=\{1,2,3,4\}$and $B=\{5,7,9\}$

$$\begin{gathered} 1.A\times B=B\times A \\ 2.n(A\times B)=n(B\times A) \end{gathered}$$

1. Statement-1 is true.

2. Statement-2 is true

3. Both are true.

4. Both are false.

Q. How many zeros in 100! ?

Q.

The product of three consecutive numbers is always divisible by 6. Verify this statement with the help of some examples.

Q. Seven athletes are participating in a race. If there are three prizes $\text{Gold, Silver}$ and $\text{Bronze}$, 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.

In a circus, there are ten cages for accommodating ten animals. Out of these four cages are so small that five out of $10$ animals cannot enter them. In how many ways will it be possible to accommodate ten animals in these ten cages?

1. $66400$

2. $86400$

3. $96400$

4. None of these

Q. Find the number of zeros at the end of $100!$

Q.

Compute :
(i) $\frac{30!}{28!}$ (ii) $\frac{11!-10!}{9!}$
(iii) L.C.M. (6!, 7!, 8!)