Combinations practice · Mixed difficulty

The total number of all proper factors of 75600 is

If ${\;^n}{C_{n - r}} + 3.{\;^n}{C_{n - r + 1}} + 3.{\;^n}{C_{n - r + 2}} + {\;^n}{C_{n - r + 3}} = {\;^x}{C_r},$ then $x =$

At an election, three wards of a town are can vassesed by 3,4 and 5 men respectively. If 20 men volunteer, then the number ways can they be allotted to the different wards is

If $\frac{2}{{9!}} + \frac{2}{{3!7!}} + \frac{1}{{5!5!}} = \frac{{{2^a}}}{{b!}},$ where $a,b, \in N$, then the ordered pair $\left( {a,b} \right)$ is

The number of ways of selecting 4 letters from 3 'a's, 4 'b's and 5 'c's is

The maximum number of points into which 4 circles and 4 straight lines intersect is

The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines, is

Everybody in a room shakes hands with everybody else. The total number of hand shakes is 66. The total number of persons in the room is

The number of ways in which 52 cards be formed into 4 groups of 13 cards each is

A car will hold 2 in the front seat and 1 in the rear seat. If among 6 persons 2 can drive, then number of ways in which the car can be filled, is

There are three men and seven women taking a dance class. Number of different ways in which each man is paired with a woman partner, and the four remaining women are paired into two pairs each of two is

A bag contains 3 black, 4 white and 2 red balls, all the balls being different. The number of selections of at most 6 balls containing balls of all the colours, is

If ${ }^{{n}} P_{{r}}=1680$ and ${ }^{{n}} C_{{r}}=70$, then $n$ is equal to

The number of ways in which 9 persons can be divided into three equal groups is

Consider the following statements :
The product of $r$ consecutive natural numbers is always divisible by $r$.
The total number of proper positive divisors of 115500 is 94
A pack of 52 cards can be divided equally among four players order in $\frac{{52!}}{{{{\left( {13!} \right)}^4}}}$ ways.
Which of the statement given above is/are correct?

If a polygon of $n$ sided has 275 diaginals, then $n$ is equal to

In an examination of 9 papers a candidate has to pass in more papers, then the number of papers in which he fails in order to be successful. The number of ways in which he can be unsuccessful, is

A crocodile is known to have not more than 68 teeth the total number of crocodiles with different set of teeth is

The number of ordered triplets of positives integers which are solutions of the equations of the equation $z + y + z = 100$, is

If ${\;^{n - 1}}{C_3} + {\;^{n - 1}}{C_4} > {\;^n}{P_3},$ then

A father with 8 children takes them 3 at a time to the zoological gardens, as often as he can without taking the same 3 children together more than once. The number of times he will go the garden, is

The value of$\,\,$ ${ }^{50} C_4+\sum_{r=1}^6{ }^{56-r} C_3$ is

The number of ways in which we can get at least 6 successive tails when a coin is tossed 10 times in a row, is

$\sum\limits_{k = 0}^{20} {{(^{20}}{C_k}} {)^2}$ is equal to

A committee of 12 is to be formed from 9 women and 8 men in which at least 5 women have to be included in a committee. Then the number of committees in which the women are in majority and men are in majority are respectively

If $\sum\limits_{k = 1}^{31} {\left( {^{31}{C_k}} \right)} \left( {^{31}{C_{k - 1}}} \right) - \sum\limits_{k = 1}^{30} {\left( {^{30}{C_k}} \right)} \left( {^{30}{C_{k - 1}}} \right) = \frac{{\alpha \left( {60!} \right)}}{{\left( {30!} \right)\left( {31!} \right)}},$ where $\alpha \in R,$ then the value of $16\alpha$ is equal to

The number of divisors of 9600 including 1 and 9600 are

These are 12 volleyball players in a college, out of which a team of 9 players is to be formed. If the captain always remains the same, then in how many ways can the team be formed?

The value of ${\;^{35}}{C_8} + \sum \limits_{r = 1}^7 {\;^{42 - r}}{C_7} + \sum \limits_{s = 1}^5 {\;^{47 - s}}{C_{40 - s}}$, is

For a game in which two partner oppose to other partner 8 men are available. If every possible pair must play with every other pair the number of games played is