Combinations practice · Easy

The number of straight lines can be formed out of 10 points of which 7 are collinear

The total number of ways in which 11 identical apples can be distributed among 6 children is

All possible two-factor products are formed from the no numbers 1,2,…,100. The number of factors out of the total obtained which are multiple of 3 is

A polygon has 44 diagonals, then the number of its sides are

The number of triplets $(x, y, z)$ where $x, y, z$ are distinct non negative integers satisfying $x+y+z=15$, is

Consider the following statements:

Statement I: The number of diagonals of $n-\text{sided}$ polygon is ${ }^n C_2-n$.

Statement II: A polygon has 44 diagonals. The number of its sides are 10.

Choose the correct option from the choices given below.

If ${ }^{10} {C}_x={ }^{10} {C}_{x+4}$, then the value of $x$ is

These are $n$ distinct points on the circumference of a circle. The number of pentagons that can be formed with these points as vertices is equal to the number of possible triangles. Then, the value of $n$ is

In a chess tournament where the participants were to play one game with one another, two players fell ill having played 6 games each, without playing among themselves. If the total number of games is 117, then the number of participants at the beginning was

If ${S_n} = \sum \limits_{r = 0}^n \frac{1}{{{\;^n}{C_r}}}$ and ${t_n} = \sum \limits_{r = 0}^n \frac{r}{{{\;^n}{C_r}}}$ then $\frac{{{t_n}}}{{{s_n}}}$ is equal to

The coefficient of ${x^{18}}$ in the product $\left( {1 + x} \right){\left( {1 - x} \right)^{10}}{\left( {1 + x + {x^2}} \right)^9}$ is

Let y be element of the set $A=\{1,2,3,5,6,10,15$, 30 $\}$ and $x_1, x_2, x_3$ be integers such that $x_1 x_2 x_3=y$, the number of positive integral solutions of $x_1 x_2 x_3=y$, is

Let ${l_1}$ and ${l_2}$ be two lines intersecting at $P$. If ${A_1},\;{B_1},\;{C_1}$ are points on ${l_1}$ and ${A_2},\;{B_2},\;{C_2},\;{D_2},\;{E_2}$ are points on ${l_2}$ and if none of these coincides with $P$, then the number of triangles formed by these eight points, is

A committee of at least three members is to be formed from a group of 6 boys and 6 girls such that it always has a boy and a girl. The number of ways to form such a committee is

The maximum number of points of intersection of 8 circles is

The number of ways in which any four letters can be selected from the word ‘CORGOO’ is

The total number of integral solutions of $x y z=24$ is

There are five different green dyes, four different blue dyes and three different red dyes. The total number of combinations of dyes that can be chosen taking at least one green and one blue dye is

Let ${T_n}$ be the number of all possible triangles formed by joining vertices of an n-sided regular polygon. If ${T_{n + 1}} - {T_n} = 10,$ then the value of $n$ is

If ${ }^{2 n} C_{3}:{ }^{n} C_{3}=10: 1$, then the ratio $\left(n^{2}+3 n\right):\left(n^{2}-\right.$ $3 n+4)$ is

The total number of ways in which a person can put 8 different rings in the fingers of his right hand is equal to

If $\frac{1}{{{\;^4}{C_n}}} = \frac{1}{{{\;^5}{C_n}}} + \frac{1}{{{\;^6}{C_n}}}$, then $n$ is equal to

There are 18 points in a plane such that no three of them are in the same line except five points which are collinear. The number of triangles formed by these points is

There are $n$-points $(n > 2)$ in each of two parallel lines. Every point on one line is joined to every point on the other line by a line segment drawn within the lines. The number of points (between the lines) in which these segments intersect is

The number of diagonals that can be drawn in a polygon of 15 sides, is

If ${ }^{{n}} C_r:{ }^{{n}} C_{r+1}:{ }^n C_{r+2}=1: 2: 3$, then $r$ is equal to

The sum $\sum\limits_{0 \leqslant i} {\sum\limits_{ \leqslant j \leqslant 10} {\left( {^{10}{C_j}} \right)\left( {^j{C_i}} \right)} }$ is equal to

The number of ways in which a mixed double game can be arranged from amongst 9 married couples if no husband and wife play in the same game is

The number of non-negative integral solutions to the system of equations $x+y+z+u+t=20$ and $x+y+z=5$ is-

The value of $\left( {{}^{21}{C_1} - {}^{10}{C_1}} \right) + \left( {{}^{21}{C_2} - {}^{10}{C_2}} \right) + \left( {{}^{21}{C_3} - {}^{10}{C_3}} \right) + \left( {{}^{21}{C_4} - {}^{10}{C_4}} \right) + ... + \left( {{}^{21}{C_{10}} - {}^{10}{C_{10}}} \right)$ is