Combinations practice · Mixed difficulty

Determine $n$ if ${ }^{2 n} {C}_3:{ }^n {C}_2=12: 1$

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

Consider the following statements.

I. ${ }^n C_r+2 \, { }^n C_{r-1}+{ }^n C_{r-2}={ }^{n+2} C_r$

II. ${ }^n C_p={ }^n C_q \Rightarrow p=q$ or $p+q=n$

Choose the correct option.

The number of ways of choosing 10 objects out of 31 objects of which 10 are identical and the remaining 21 are distinct, 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 $\left({ }^{30} C_{1}\right)^{2}+2\left({ }^{30} C_{2}\right)^{2}+3\left({ }^{30} C_{3}\right)^{2}+\ldots+30\left({ }^{30} C_{30}\right)^{2}$ $=\frac{\alpha 60 !}{(30 !)^{2}}$ then $\alpha$ is equal to :

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

${ }^m C_{r+1}+\sum\limits_{{k=m}}^n{ }^k C_r$ is equal to

Statement-1: The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is empty is ${ }^{9} C_{3}$.
Statement-2: The number of ways of choosing any 3 places from 9 different places is ${ }^{9} C_{3}$.

If $m$ denotes the number of 5 digit number if each successive digits are in their descending order magnitude and ${n}$ is the corresponding figure when the digit are in their ascending order of magnitude

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 different ways in which five alike dashes and eight alike dots can be arranged using only seven of these 'dashes' and 'dots' is equal to

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