Combinations practice · Mixed difficulty

If $20 \%$ of three subsets (i.e., subsets containing exactly three elements ) of the set $A=\left\{a_1, a_2 \cdots a_n\right\}$ contain $a_1$, then the value of $n$ is

A lady gives a dinner party to 5 guests to be selected from nine friends. The number of ways of forming the party of 5 , given that two of the friends will not attend the party together, is

Pyramid has a square base. Vertices are to be selected from the set $\{1,2,3,4,5\}$ such that no two adjacent vertices take same number, the number of ways is

Let there are 6 shirts of different colours and 6 trousers of same colours as that of shirts. If the number of ways in which these can be put on by 5 men such that no men wear the shirt and the trouser of the same colour is $(k) 6!,$ then find the value of $k$.

The number of $2 \times 2$ matrices having elements 0 and 1, is

Statement-1 The number of ways distributing 10 identical balls in 4 distinct boxes such that no box is empty, is ${ }^9 C_3$.
Statement-2 The number of ways of choosing any 3 places from 9 different places is ${ }^9 C_3$.

Three straight lines ${L_1},\;{L_2},\;{L_3}$ are parallel and lie in the same plane. A total of m points are taken on ${L_1},\;n$ points on ${L_2},\;k$ points on ${L_3}$. The maximum number of triangles formed with vertices at these points are

If $a, b, c$ are odd positive integers, then number of integral solutions of $a+b+c=13$, is

The number of triangles which can be formed by using the vertices of a regular polygon of $\left( {n + 3} \right)$ sides is 220. Then $n$ is equal to

At an election there are five candidates and three members are to be elected, and a voter may vote for any number of candidates not greater than the number to be elected. The number of ways in which the person can vote is

The number of diagonals that can be drawn by joining the vertices of an octagon is

The number of positive integer solutions of $a+b+c=60$, where $a$ is a factor of $b$ and $c$, is

If $\sum_{i=1}^{20}\left(\frac{{ }^{20} \mathrm{C}_{i-1}}{{ }^{20} \mathrm{C}_{i}+{ }^{20} \mathrm{C}_{i-1}}\right)^{3}=\frac{k}{21}$, then $k$ equals:

The number of ways of dividing 52 cards amongst four players so that three players have 17 cards each and the fourth players just one card, is

Let ${T_n}$ denote the number of triangles which can be formed using the vertices of a regular polygon of $n$ sides. If ${T_{n + 1}} - {T_n} = 21$, then $n$ equals

The number of subsets of $\left\{ {1,{\text{ }}2,{\text{ }}3,{\text{ }} \ldots ,{\text{ }}9} \right\}$ containing at least one odd number is

A student is allowed to select at most $n$ books from a collection of $\left( {2n + 1} \right)$ books. If the total number of ways in which he can select one book is 63, then the value of $n$ is equal to

In a shop, there are five types of ice-creams available. A child buys six ice-creams.
Statement-1 The number of different ways the child can buy the six ice-creams is ${ }^{10} \mathrm{C}_4$.
Statement-2 The number of different ways the child can buy six ice-creams is equal to the number of different ways to arranging $6 A^{\prime} s$ and $4 B^{\prime} s$ in a row.

Out of 11 consecutive natural numbers if three numbers are selected at random (without repetition), then the probability that they are in AP with positive common difference, is

Number of triangles that can be formed joining the angular points of decagon is

The number of ways in which 13 gold coins can be distributed among three persons such that each one gets at least 2 gold coins is

If ${\;^n}{C_r}$ denotes the number of combinations of $n$ things takes $r$ at a time, then the expression ${\;^n}{C_{r + 1}} + {\;^n}{C_{r - 1}} + 2 \times {\;^n}{C_r}$, equals

On the occasion of Diwali festival each student of a class sends greeting cards to the others. If there are 20 students in the lass, then the total number of greeting cards exchanged by the students is

Team A consists of 7 boys and n girls and team B has 4 boys and 6 girls. If a total of 52 single matches can be arranged between these two teams when a boy plays against a boy and a girl plays against a girl, then n=

There are two parallel lines one having 10 point and other having 5 points. The number of triangle formed with vertice as these points

The number of divisors of the number of 38808 (excluding 1 and the number itself)is

The number of ways in which four letters can be selected from the word degree is

If $a_n=\sum\limits_{{r=0}}^n \frac{1}{{ }^n C_r}$, then $\sum\limits_{{r=0}}^n \frac{r}{{ }^n C_r}$ equals

In a plane, there are 37 straight lines, of which 13 pass through the point $A$ and 11 pass through the point $B$. Besides, no three lines pass through one point, no line passes through both points $A$ and $B$, and no two are parallel. Then the number of intersection points the lines have is equal to

If ${\;^{n - 1}}{C_6} + {\;^{n - 1}}{C_7} > {\;^n}{C_6},$ then