Combinations practice · Mixed difficulty

If $\frac{1}{{^4{C_n}}} = \frac{1}{{^5{C_n}}} + \frac{1}{{^6{C_n}}},$ then value of $n$ is

The number of ways in which an examiner can assign 30 marks to 8 questions, giving not less than 2 marks to any question, is :
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A box contains 5 pairs of shoes. If 4 shoes are selected, then the number of ways in which exactly one pair of shoes obtained is

A committee of 7 has to be formed from 9 boys and 4 girls.

I. In 504 ways, this can be done, when the committee consists of exactly 3 girls.

II. In 588 ways, this can be done, when the committee consists of at least 3 girls.

Choose the correct option.

Number if divisors of the form $\left( {4n + 2} \right),n \ge 0$ of the integer 240 is

The number of integral solution of the inequality $4 < x_1+x_2+x_3+x_4 < 9$ where $x_1, x_2, x_3, x_4 \geq 0$

7 relatives of a man comprise 4 ladies and 3 gentlemen his wife has also 7 relatives, 3 of them are ladies and 4 gentlemen. In how many ways can they invite a dinner party of 3 ladies and 3 gentlemen so that there are 3 of man’s relative and 3 of the wife’s relative?

If ${\;^{16}}{C_r} = {\;^{16}}{C_{r + 1}}$, then the value of ${\;^r}{P_{r - 3}}$ is

A committee of 5 is to be formed from 9 ladies and 8 men. If the committee commands a lady majority, then the number of ways this can be done is

The number of selecting at least 4 candidates from 8 candidates is

There are 12 pairs of shoes in a box. Then the possible number of ways of picking 7 shoes so that there are exactly two pairs of shoes are

If ${\;^n}{C_{r - 1}} = 36,\;{\;^n}{C_r} = 84$ and ${\;^n}{C_{r + 1}} = 126$, then the value of $r$ is

If $a, b$ and $r$ are positive integers, then value of ${ }^a C_r+{ }^a C_{r-1}{ }^b C_1+{ }^a C_{r-2}{ }^b C_2+\ldots+{ }^b C_r \text { is}$

From a company of 20 soldiers any 5 are placed on guard, each batch to watch 5 hours. The total number of hours watched by different batches selected is

If $r > p > q,$ the number of different selections of $p + q$ thing taking r at a time, where p things are identical and $q$ other things are identical, is

Eight persons are to be transported from city $A$ to city $B$ in three cars of different makes. If each car can accommodate at most three persons, then the number of ways, in which they can be transported, is

The product of $n$ consecutive natural number is always divisible by

There are three pigeon holes marked $M, P, C.$ The number of ways in which we can put 12 letters so that 6 of them are in $M,$ 4 are in $P$ and 2 are in $C$ is

$\sum\limits_{{0 \leq i \leq j \leq 10}} \sum^{10} C_i \cdot{ }^j C_i=$

If $n$ is an integer with $0 \leq n \leq 11,$ then the minimum value of $n !(11-n) !$ is attained, when a value of $n$ is

The total number of integral solution for $x, y, z$ such that $x y z=24$, is

If $a,\,b$ and $c$ are the greatest values of $^{19}{C_p}{\,^{20}}{C_q}$ and $^{21}{C_r}$ respectively, then

$\sum\limits_{r=0}^{n}{\left( \frac{r+3}{r+1} \right)} n_{c_r}=\frac{7 \cdot\left(2^{10}\right)-2}{10}$ then $n=$

The greatest number of points of intersection of $n$ circles and $m$ straight lines is

The number of straight lines can be formed out of 10 points of which 7 are collinear

The number of ways choosing 10 balls from infinite white, red blue and green balls is

The total number of ways in which 11 identical apples can be distributed among 6 children is

A student is allowed to select atmost $n$ books from a collection of $(2 n+1)$ books. If the total number of ways in which he can select atleast one book is 225, then the value of $n$ is

All possible two-factor products are formed from the no numbers 1,2,…,100. The number of factors out of the total obtained which are multiple of 3 is

A polygon has 44 diagonals, then the number of its sides are