Combinations practice · Easy

There were two women participating in a chess tournament. Every participant played two games with the other participants. The number of games that the men played between themselves proved to exceed by 66 the number of games that the men played with the women. The number of participants 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

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, the number of intersection points the lines have, is

The straight lines $I_1, I_2, I_3$ are parallel and lie in the same plane. A total number of $m$ points on $I_1 ; n$ points on $I_2 ; k$ points on $I_3$, the maximum number of triangles formed with vertices at these points is

There are $n$ points in a plane, no three of which are collinear except ' $p$ ' points all of which are on a line. How many
(a) straight lines can be formed and
(b) triangles can be formed out of these $n$ points?

The number of ways of selecting 4 cards of an ordinary pack of playing cards so that exactly 3 of them are of the same denomination is

The number of diagonals of a polygon of 15 sides is

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

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

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

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

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

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

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

A person is permitted to select at least one and at most $n$ coins from a collection of $2n + 1$ (distinct) coins. If the total number of ways in which he can select coins is 255, then $n$ equals