If  ${{}^{20}C}_{r+2}={{}^{20}C}_{2r-3}$, find ${{}^{12}C}_r$.

Important Questions on Counting Principles

EASY

Aptitude>Arithmetic>Counting Principles>Combinations

There are $m$ men and two women participating in a chess tournament. Each participant plays two games with every other participant. If the number of games played by the men between themselves exceeds the number of games played between the men and the women by $84$, then the value of $m$ is :

EASY

Aptitude>Arithmetic>Counting Principles>Combinations

Two women and some men participated in a chess tournament in which every participant played two games with each of the other participants. If the number of games that the men played between them-selves exceeds the number of games that the men played with the women by$66$, then the number of men who participated in the tournament lies in the interval

HARD

Aptitude>Arithmetic>Counting Principles>Combinations

The value of $\sum _{r=1}^{15}{r}^2\left(\frac{{}^{15}{C}_r}{{}^{15}{C}_{r-1}}\right)$ is equal to:

EASY

Aptitude>Arithmetic>Counting Principles>Combinations

Consider three boxes, each containing $10$ balls labelled $1, 2, \dots ., 10$. Suppose one ball is randomly drawn from each of the boxes. Denote by $n_i$, the label of the ball drawn from the $i^{\mathrm{t}\mathrm{h}}$ box, $\left(i=1, 2, 3\right)$. Then, the number of ways in which the balls can be chosen such that $n_1<n_2<n_3$ is :

HARD

Aptitude>Arithmetic>Counting Principles>Combinations

Let $A=\left\{x_1,x_2,\dots ,x_7\right\} \text{and} B=\left\{y_1,y_2,y_3\right\}$ be two sets containing seven and three distinct elements respectively. Then the total number of functions $f:A\to B$ that are onto, if there exist exactly three elements $x$ in $A$ such that $f\left(x\right)=y_2,$ is equal to:

MEDIUM

Aptitude>Arithmetic>Counting Principles>Combinations

If the four letter words (need not be meaningful) are to be formed using the letters from the word "MEDITERRANEAN" such that the first letter is $\mathrm{R}$ and the fourth letter is $\mathrm{E}$, then the total number of all such words is :

MEDIUM

Aptitude>Arithmetic>Counting Principles>Combinations

If $\frac{{}^{n+2}{C}_6}{{}^{n-2}{P}_2}=11$, then $n$ satisfies the equation:

MEDIUM

Aptitude>Arithmetic>Counting Principles>Combinations

A committee of $11$ member is to be formed from $8$ males and $5$ females. If $m$ is the number of ways the committee is formed with at least $6$ males and $n$ is the number of ways the committee is formed with at least $3$ females, then

MEDIUM

Aptitude>Arithmetic>Counting Principles>Combinations

The number of numbers between $2,000$ and $5,000$ that can be formed with the digits $0, 1, 2, 3, 4$ (repetition of digits is not allowed) and are multiple of $3$ is

MEDIUM

Aptitude>Arithmetic>Counting Principles>Combinations

A man $X$ has $7$ friends, $4$ of them are ladies and $3$ are men. His wife $Y$ also has $7$ friends, $3$ of them are ladies and $4$ are men. Assume $X$ and $Y$ have no common friends. Then the total number of ways in which $X$ and $Y$ together can throw a party inviting $3$ ladies and $3$ men, so that $3$ friends of each of $X$ and $Y$ are in this party is:

HARD

Aptitude>Arithmetic>Counting Principles>Combinations

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

HARD

Aptitude>Arithmetic>Counting Principles>Combinations

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 :

MEDIUM

Aptitude>Arithmetic>Counting Principles>Combinations

The number of integers greater than $6000$ that can be formed, using the digits $3, 5, 6, 7$ and $8$, without repetition is 

MEDIUM

Aptitude>Arithmetic>Counting Principles>Combinations

The number of natural numbers less than $7000$ which can be formed by using the digits $0, 1, 3, 7, 9$ (repetition of digits allowed) is equal to:

MEDIUM

Aptitude>Arithmetic>Counting Principles>Combinations

A group of students comprises of $5$ boys and $n$ girls. If the number of ways, in which a team of $3$ students can randomly be selected from this group such that there is at least one boy and at least one girl in each team, is $1750,$ then $n$ is equal to

EASY

Aptitude>Arithmetic>Counting Principles>Combinations

The Number of ways of choosing $10$ objects out of $31$ objects of which $10$ are identical and the remaining $21$ are distinct, is:

MEDIUM

Aptitude>Arithmetic>Counting Principles>Combinations

Suppose that $20$ pillars of the same height have been erected along the boundary of circular stadium. If the top of each pillar has been connected by beams with the top of all its non-adjacent pillars, then the total number of beams is:

MEDIUM

Aptitude>Arithmetic>Counting Principles>Combinations

From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on the shelf so that the dictionary is always in the middle. Then the number of such arrangement is

EASY

Aptitude>Arithmetic>Counting Principles>Combinations

The number of ways of selecting $15$ teams from $15$ men and $15$ women, such that each team consists of a man and a woman is

MEDIUM

Aptitude>Arithmetic>Counting Principles>Combinations

A debate club consists of $6$girls and $4$ boys. A team of $4$ members is to be selected from this club including the selection of a captain (from among these $4$ members) for the team. If the team has to include at most one boy, then the number of ways of selecting the team is