Permutation and combination practice · Mixed difficulty

If a man and his wife enter in a bus, in which five seats are vacant, then the number of different ways in which they can be seated, is

The number of ways in which 5 boys and 3 girls can be seated in a row so that each girl is between two boys is

${ }^1 \mathrm{P}_1+2 .{ }^2 \mathrm{P}_2+3 .{ }^3 \mathrm{P}_3+\cdots+n .{ }^n \mathrm{P}_n$ is equal to

The number of different words that can be formed from the letters of the word ‘PENCIL’ so that no two vowels are together, is

There are 10 persons among whom two are brother. The total number of ways in which these persons can be seated around a round table so that exactly one person sit between the brothers, is equal to

Let $p$ be a prime number such that $p \geq 11$. Let $n=p !+1$. The number of primes in the list $n+1, n+2, n+3, \ldots n+p-1$, is

If there are 12 persons in a party, and if each of them shakes hands with each other, then number of handshakes happen in the party is

The pages of a book are numbered 1 to $\mathrm{n}$. When page numbers of the book were added, one of the page number was mistakenly added twice \& hence resulting is a incorrect sum 1986 . What was the number of page that was added twice :

The number of words that can be formed from the letters of the word DAUGHTER so that the vowels always come together, is

The sum of all the four-digit numbers that can be formed using all the digits $2,1,2,3$ is equal to____

The number of ways of painting the faces of a cube with six different colours is

The number of ways in which the letters of the word 'SUCCESSFUL' be arranged such that no two S's will come together is

Determine $n$, if ${ }^{\text {2n}} C_2:{ }^n C_2=9: 2$.

${X}$ has four friends ${A}, {B}, {C}, {D}$ he wants to invite one or more of them at dinner. If the invited persons(s) $\& \,X$ are sitting around a circular table then the arrangement in which $C\,\&\,D$ (if both are invited) are sitting together is

The total number of ways of selecting two numbers from the set $\{1,2,3,4, \ldots, 3 n\}$, so that their sum of divisible by 3, is

The total number of functions ' $f$ ' from the set $\{1,2,3\}$ into the set $\{1,2,3,4,5\}$ such that $f(i) \leqslant f(j),\forall i < j$, is

Let $a$ be a factor of 120, then the number of positive integral solutions of $x_1 x_2 x_3=a$ is

The sum $1\left( {1!} \right) + 2\left( {2\;!} \right) + 3\left( {3!} \right) + ... + n\left( {n!} \right)$ equal to

In a group of boys, two boys are brothers and in this group, 6 more boys are there. In how many ways, they can sit if the brothers are not to sit alongwith each other :

A library has a copies of one book, $b$ copies of each of two books, $c$ copies of each of three books and single copies of d books. The total number of ways in which these book can be distributed, is

Let $S_n=\sum\limits_{{r=1}}^n r!$. If $T_n=S_n-7\left[\frac{S_n}{7}\right]$, where $[x]$ denotes greatest integer $\leq x$, then $T_{100}$ is equal to

Total number of ways in which 15 identical blankets can be distributed among 4 persons, so that each of them gets atleast two blankets, is

If $\mathrm{k}$ be the number of 3 digit natural numbers, having sum of their digits at least 10 , then find $\frac{k-35}{100}$.

If ${\;^{2n + 1}}{P_{n - 1}}:{\;^{2n - 1}}{P_n}:3:5$, then the value of $n$ is equal to

If ${ }^{15} C_{{r}+1}:{ }^{15} C_{3 {r}}=3: 11$, then value of $r$ is:

There are 10 points in a plane, of which no 3 points are collinear and 4 points are concyclic. The number of different circles that can be drawn through atleast 3 of these points, is

The lock of a safe consists of five discs each of which features the digits 0, 1, 2, … ,9.The safe can be opened by dialing a special combination of the digits. The number of days sufficient enough to open the safe. If the work day lasts 13 h and 5 s are needed to dial one combination of digits is

In a class tournament, all participants were to play different games with one another. Two players fell ill after having played three games each. If the total number of games played in the tournament is equal to 84 , the total number of participants in the beginning was equal to

The product of $r$ consecutive positive integers is divisible by

In a steamer there are stalls for 12 animals and there are cows, horses and calves (not less than 12 of each) ready to be shipped. The total number of ways in which the ship load can be made is