Combination

Find the total number of combinations of three letters, J, R, T taken $1$, $2$, $3$ at a time.

The number of ways can $3$ nurses and $4$ engineers be seated on a bench with the nurses seated together is?

In how many ways can a committee of three consisting of two chemical engineers and one mechanical engineer can be formed from four chemical engineers and three mechanical engineers?

How many committees can be formed by choosing $4$ men from an organisation of a membership of $15$ men?

There are four balls of different colours. Two balls at a time are taken and arranged any way. How many such combinations are possible?

If ${}^{1000}c_{98}={}^{999}c_{97}+{}^{x}c_{901}$ then the value of x will be

There are $13$ teams in a tournament. Each team is to play with each other only once. What is the minimum number of days can they all play without any team playing more than one game in any day?

Four different coloured flags can be hung in a row to make coded signal. How many signals can be made if a signal consists of the display of one or more flags?

In how many ways can you invite one or more of your five friends in a party?

How many different ways can $5$ boys and $5$ girls form a circle with boys and girls alternate?

A semiconductor company will hire $7$ men and $4$ women. In how many ways can the company choose from $9$ men and $6$ women who qualified for the position?

There are five main roads between the cities $\mathrm{A}$ and $\mathrm{B}$, and four between $\mathrm{B}$ and $\mathrm{C}$. In how many ways can a person drive from  $\mathrm{A}$ to $\mathrm{C}$ and return, going through $\mathrm{B}$ on both trips without driving on the same road twice?

An examination paper consists of$12$ questions divided into parts$A$ &$B$. Part-$A$ contains$7$ questions & Part-$B$ contains$5$ questions. A candidate is required to attempt$8$ questions selecting$3$ at least from each part. In how many maximum ways can the candidate select the questions?

The total number of paths from the origin to the point $\left(9,9\right)$ in the cartesian plane which never pass through $\left(5,5\right)$, where paths consisting only of steps going $1$ unit North or $1$ unit East, is

The number of positive divisors of $10!$ are

Let the number of elements in sets $A$ and $B$ be five and two respectively. Then the number of subsets of $A\times B$ each having  at least $3$ and at most $6$ elements is

If ${}^{2n}C_3:{}^{n}C_3=10:1,$ then the ratio $\left(n^2+3n\right):\left(n^2-3n+4\right)$ is

If we have a ATM pin of $4$ digit.The sum of first two digit is equal to the sum of last two digits and the greatest integer used is $7$, then number of trials used to get the pin if all digits are different