Combination of Atleast One Thing from n Things When All Are Not Different

Q. $P,Q,R,S,T,U,V$ and $W$ are sitting round the circle and are facing the centre :
$1.P$ is second to the right of $T$ who is the neighbour of $R$ and $V.$
$2.S$ is not the neighbour of $P.$
$3.V$ is the neighbour of $U.$
$4.Q$ is not between $S$ and $W.$
$5.W$ is not between $U$ and $S.$
Which two of the following are not neighbours?

1. $RV$
2. $UV$
3. $RP$
4. $SW$

Q. How many $4-$digit numbers can be formed by using the digit $1$ to $9$ if repetition of digits is not allowed?

Q. Find $r$, if $5{}^4{P}_r=6{}^5{P}_{r-1}$

Q. If ${}^n{C}_9={}^n{C}_8,$ find ${}^n{C}_{17}$

Q. Let $n$ be the number of ways in which $5$ boys and $5$ girls can stand in a queue in such a way that all the girls stand consecutively in the queue. Let $m$ be the number of ways in which $5$ boys and $5$ girls can stand in a queue in such a way that exactly four girls stand consecutively in the queue. Then the value of $\frac{m}{n}$ is

Q. The total number of numbers, lying between $100$ and $1000$ that can be formed with the digits $1,2,3,4,5,$ if the repetition of digits is not allowed and numbers are divisible by either $3$ or $5$, is

Q. Find the number of $4-$ digit number that can be formed using the digits $1,2,3,4,5$ if no digit is repeated. How many of these will be even?

Q. How many $3-$digit numbers can be formed from the digits $1,2,3,4$ and $5$ assuming that
$\left(i\right)$ Repetition of the digits is allowed.
$\left(ii\right)$ Repetition of the digits is not allowed.

Q.

How many $3$-digit natural numbers are completely divisible by $5$?

Q.

In how many ways can a pack of 52 cards be divided in 4 sets, three of them having 17 cards each and fourth just 1 card?

1. 

2. 

3. 

4. 

Q. The number of ways in which $12$ different books can be put in three shelves with four on each shelf is

1. $\frac{12!}{(3!)(4!{)}^3}$
2. $\frac{12!}{(4!{)}^3}$
3. $\frac{12!}{(3!{)}^{3}4!}$
4. none of these

Q. If seven-digit number made up of all distinct digits $8,7,6,4,2,x$ and $y$ is divisble by $3$, then

1. Maximum value of $x-y$ is $9$
2. maximum value of $x+y$ is $12$
3. $\sum (x+y)=60$
4. minimum value of $x-y$ is $3$

Q.

Insert commas in the following numbers as per the International system of Numeration

a) $54534423$

Q. In how many ways can the letters of the word $PERMUTATIONS$ be arranged if the
$\left(i\right)$ Words starts with $P$ and end with $S$.
$\left(ii\right)$ Vowels are all together
$\left(iii\right)$ There are always $4$ letters between $P$ and $S$ ?

Q. If a bag has $3$ alike mangoes, $4$ alike apples and $2$ alike oranges, then the correct option(s) is/are?

1. Total number of ways of selecting atleast $1$ of each kind of fruit is $24$
2. Total number of ways of selecting atleast $1$ fruit is $60$
3. Total number of ways of selecting atleast $1$ of each kind of fruits is $23$
4. Total number of ways of selecting atleast $1$ fruit is $59$

Q. How many $3-$ digit numbers can be formed by using the digits $1$ to $9$ if no digit is repeated?

Q. Let $n$ be the number of ways in which $5$ boys and $5$ girls can stand in a queue in such a way that all the girls stand consecutively in the queue. Let $m$ be the number of ways in which $5$ boys and $5$ girls can stand in a queue in such a way that exactly four girls stand consecutively in the queue. Then the value of $\frac{m}{n}$ is

Q. The number of divisors of $2^2{.3}^3{.5}^3{.7}^5$ of the form 2n+1, n $ϵN$ is

1. 96
2. 95
3. 94
4. 924

Q. If the number of three digit numbers having only two consecutive digits identical is $N,$ then the value of $(N/2{)}^{1/2}$ is

Q. Total number of $4$ digit numbers that can be formed using the digits $1,2,5,6,7$ and that are divisible by $4$ is

Q. How do you simplify $3^4\cdot 3^5$ ?

Q. There are $(n+1)$ white and $(n+1)$ black balls, each set numbered from $1$ to $n+1$. The number of ways in which the balls can be arranged in a row so that the adjacent balls are of different colours is

1. $(2n+2)!$
2. $\left[\right(n+1)!{]}^2$
3. $2\left[\right(n+1)!{]}^2$
4. $(n+1)!$

Q. If $3^{37}=80\lambda +k,where\lambda ϵN,$then 'k' is

1. 78
2. 3
3. 2
4. 9

Q. The number of $4$-digit numbers that can be formed using the digits $1,2,3,4,5$ that are divisible by $6$ when repetition is allowed.

Q.

Insert commas in the correct places and write the given below number in words according to Indian and International systems of numeration.

$56712345$

Q. The sum of the digit of a two digit number is $15$ and the difference between the digits is $3$. What the two digit number?

1. $78$
2. $69$
3. $90$
4. $86$

Q. If $nϵN,7^{2n}+2^{3n-3}$ is always divisible by

1. 35
2. 25
3. None of these
4. 45

Q. How many numbers less than 2000 that can be formed using the digits $1,2,3,4$ if repetition is allowed.

Q. The number of ways in which $12$ different books can be put in three shelves with four on each shelf is

1. $\frac{12!}{(4!{)}^3}$
2. $\frac{12!}{(3!)(4!{)}^3}$
3. $\frac{12!}{(3!{)}^{3}4!}$
4. none of these

Q.

if a, b, c $ϵ$ R, a, b, c $\ne$ 0, $\sum$ $a^2$ = $\sum$ab, then a, b, c are in

1. A.P

2. G.P

3. H.P

4. A.G.P