Combinations practice · Easy

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 $A B C D$ 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 $P\left( {n\;,r} \right) = 1680$ 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

If $n$ is even and $^n{C_0}{ < ^n}{C_1}{ < ^n}{C_2} < ....{ < ^n}{C_r}{ > ^n}{C_{r + 1}}{ > ^n}{C_{r + 2}} > ....{ > ^n}{C_{n'}}$ then, $r =$

Sum of all the odd divisors of 720 is

The number of ways of distributing 8 identical balls in 3 distinct boxes, so that none of the boxes is empty, is

A rectangle with sides $2m - 1$ and $2n - 1$ divided into squere of unit length. The number of rectangle which can be formed with sides of odd length is

In a conference of 8 persons, if each person shake hand with the other one only, then the total number of shake hands shall be

If ${a_n} = \sum \limits_{r = 0}^m \frac{1}{{{\;^n}{C_r}}},\;{\text{then}}\;\sum \limits_{r = 0}^n \frac{r}{{{\;^n}{C_r}}}$ equals

The number of ways in which a committee can be formed of 5 members from 6 men and 4 women if the committee has at least one woman, is

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

The sides $AB,\;BC,\;CA$ of a triangle $ABC$ have 3,4 and 5 interior points respectively on them. The total number of triangles that can be constructed by using these points as vertices is

There is a set of $m$ parallel lines intersecting a set of another $n$ parallel lines in a plane. The number of parallelograms formed, is

The value of $\sum_{r=1}^{10} r \cdot \frac{{ }^n C_r}{{ }^n C_{r-1}}$ is equal to

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

If $\frac{{ }^{n+2} C_{6}}{{ }^{n-2} P_{2}}=11$, then $\mathrm{n}$ satisfies the equation :

The number of ways of selecting two numbers a and $b, \quad a \in\{2,4,6, \ldots .100\}$ and $b \in(1,3,5, \ldots, 99)$ such that 2 is the remainder when $a+b$ is divided by 23 is

The number of ways in which a committee of 3 women and 4 men be chosen from 8 women and 7 men if Mr. $X$ refuses to serve on the committee if $\mathrm{Mr}$. $Y$ is a member of the committee is

If $a, b, c$ and $d$ are odd natural numbers such that $a+b+c+d=20$, the number of values of the ordered quadruplet $(a, b, c, d)$ is