The number of diagonals in a octagon will be

Important Questions on Permutation and Combination

MEDIUM

Mathematics>Algebra>Permutation and Combination>Combination

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

EASY

Mathematics>Algebra>Permutation and Combination>Combination

If ${{}^{n}C}_{r-1}=36, {{}^{n}C}_r=84$ and ${{}^{n}C}_{r+1}=126$$,$ then $n=$

EASY

Mathematics>Algebra>Permutation and Combination>Combination

A password is set with $3$ distinct letters from the word LOGARITHMS. How many such passwords can be formed?

EASY

Mathematics>Algebra>Permutation and Combination>Combination

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

Mathematics>Algebra>Permutation and Combination>Combination

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:

HARD

Mathematics>Algebra>Permutation and Combination>Combination

In a tournament with five teams, each team plays against every other team exactly once. Each game is won by one of the playing teams and the winning team scores one point, while the losing team scores zero. Which of the following is NOT necessarily true?

MEDIUM

Mathematics>Algebra>Permutation and Combination>Combination

The number of diagonals of a polygon with $15$ sides is

MEDIUM

Mathematics>Algebra>Permutation and Combination>Combination

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

HARD

Mathematics>Algebra>Permutation and Combination>Combination

If in a regular polygon the number of diagonals is $54$, then the number of sides of this polygon is:

HARD

Mathematics>Algebra>Permutation and Combination>Combination

The number of noncongruent integer-sided triangles whose sides belong to the set $\{10, 11, 12,......,22\}$ is

MEDIUM

Mathematics>Algebra>Permutation and Combination>Combination

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

HARD

Mathematics>Algebra>Permutation and Combination>Combination

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 :

HARD

Mathematics>Algebra>Permutation and Combination>Combination

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

MEDIUM

Mathematics>Algebra>Permutation and Combination>Combination

The least value of a natural number $n$ such that $\left(\frac{n-1}{5}\right)+\left(\frac{n-1}{6}\right)<\left(\frac{n}{7}\right)$, where $\left(\frac{n}{r}\right)=\frac{n!}{\left(n-r\right)! r!}$ , is

HARD

Mathematics>Algebra>Permutation and Combination>Combination

Let $S=\left\{\mathrm{1,2},3,\dots .9\right\}$. For $k=\mathrm{1,2},\dots 5,$ let $N_k$ be the number of subsets of $S$, each containing five elements out of which exactly $k$ are odd. Then $N_1+N_2+N_3+N_4+N_5=$

EASY

Mathematics>Algebra>Permutation and Combination>Combination

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

Mathematics>Algebra>Permutation and Combination>Combination

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

Mathematics>Algebra>Permutation and Combination>Combination

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:

EASY

Mathematics>Algebra>Permutation and Combination>Combination

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

EASY

Mathematics>Algebra>Permutation and Combination>Combination

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 :