Combinations practice · Easy

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

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

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

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

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 ${ }^a C_r+{ }^a C_{r-1}{ }^b C_1+{ }^a C_{r-2}{ }^b C_2+\ldots+{ }^b C_r \text { is}$

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

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

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