Combinations practice · Mixed difficulty

If $\sum\limits_{r = 0}^{25} {\left\{ {{}^{50}{C_r} \cdot {}^{50 - r}{C_{25 - r}}} \right\} = K\left( {{}^{50}{C_{25}}} \right)}$. Then, $K$ is equal to

In a shop there are five types of ice-creams (each type is available unlimited). A child buys six ice-creams.
Statement-1: The number of different ways the child can buy the six ice-creams is ${ }^{10} C_5$.
Statement-2: The number of different ways the child can buy the six ice-creams is equal to the number of different ways of arranging $6 {A}^{\prime} \mathrm{s}$ and $4 {B}$ 's in a row.

The number of diagonals of a polygon of 20 sides is

In a club election the number of contestants is one more than the number of maximum candidates for which a voter can vote. If the total number of ways in which a voter can vote be 126, then the number of contestants is

At an electron, a voter may vote for any number of candidates not greater than the number to be elected. There are 10 candidates and 4 are to be elected. If a voter votes for at least one candidate, then the number of ways in which he can vote, is

Let $A$ be the set of 4 digit number ${a_1}{a_2}{a_3}{a_4},\;\mathrm{where}\;{a_1} < {a_2} < {a_3} < {a_4},$ then $n\left( A \right)$ is equal to

There are 10 points in a plane, out of these 6 are collinear. If $N$ is the number of triangles formed by joining these points, then

Given 6 different toys of green colour, 5 different toys of blue co- lour and 4 different toys of red colour. Combination of toys that can be chosen taking at least one green and one blue toys are

There are 5 points ${P}_1, {P}_2, {P}_3, {P}_4, {P}_5$ on the side ${AB}$, excluding $A$ and $B$, of a triangle $A B C$. Similarly there are 6 points ${P}_6, {P}_7, \ldots, {P}_{11}$ on the side ${BC}$ and 7 points ${P}_{12}, {P}_{13}, \ldots, {P}_{18}$ on the side ${CA}$ of the triangle. The number of triangles, that can be formed using the points ${P}_1, {P}_2, \ldots, {P}_{18}$ as vertices, is:

$p$ points are chosen on each of the three coplanar lines. The maximum number of triangles formed with vertices at these points is

A dice is rolled 4 times, the numbers appearing on the dice in different throws are listed, such that the largest number appearing in the list is not 4 , in how many ways this can be done

The number of diagonals in a octagon will be

The sum $\sum_{r=1}^{10}\left(r^{2}+1\right) \times(r !)$ is equal to :

If ${\;^n}{C_r} = {\;^n}{C_{r - 1}}{\text{\;and}}\;{\;^n}{P_r} = {\;^n}{P_{r + 1}}$, then the value of $n$ is

The number of ways in which a team of eleven players can be selected from 22 players including 2 of them and excluding 4 of them is

There are four balls of different colours and four boxes of colours same as those of the balls. The number of ways in which the balls, one in each box, could be placed such that a ball does not go to box of its own colour, is

If ${}^n{P_r} = {}^n{P_{r + 1\,\,\,}}\,\,\mathrm{and}\,\mathrm{ }{}^n{C_r} = {}^n{C_{r - 1}},$ then the value of $r$ is equal to

The number of ways can a team of six horses be selected out of a stud of 16 , so that there shall always be three out of $A, B, C, D, E, F$ but never $A D, B E$ or $C F$ together, is

If the number of 5-element subsets of the set $A=\left\{a_{1}, a_{2}, \ldots, a_{20}\right\}$ of 20 distinct elements is $k$ times the number of 5 -element subsets containing $a_{4}$, then $k$ is

If $A B C D$ is a convex quadrilateral with $3,4,5$ and 6 points, marked on sides $A B, B C, C D$ and $D A$ respectively, then the number of triangles with vertices on different sides, is

If ${ }^n C_r$ denotes the number of combinations of $n$ things taken $r$ things at a time, then the expression ${ }^n C_{r+1}+{ }^n C_{r-1}+2{ }^n C_r$ is

The number of all the odd divisors of 3600 is

In an examination a candidate has to pass in each of the papers to be successful. It the total number of ways to fail is 63 , how many papers are there in the examination?

If $P\left( {n\;,r} \right) = 1680$ and $C\left( {n,\;r} \right) = 70$, then $69n + r!$ is equal to

$\sum\limits_{{k=0}}^{6}{ }^{51-k} C_{3}$ is equal to

Ramesh has 6 friends. In how many ways can he invite one or more of them at a dinner?

A question paper is divided into two parts $A$ and $B$ and each part contain 5 questions. The number of ways in which a candidate can answer 6 questions selecting at least two questions from each part is

Total number of ways of selecting five letters from letters of the word INDEPENDENT is

The total number of proper divisors of 38808 is

There are $n$ straight lines in a plane, no two of which are parallel and no three pass through the same point. Their points of intersection are joined. Then, the number of fresh lines thus obtained is