Combinations

If ${}^{\mathrm{15}}{C}_{3r}={}^{15}{C}_{r+3}$, then the value of $r$ is

If  ${{}^{20}C}_{r+2}={{}^{20}C}_{2r-3}$, find ${{}^{12}C}_r$.

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

If total number of students is $10$ and number of students to be selected or permutated is $5$. Find the number of ways of selection and permutation and state the difference between permutation and combination.

If total number of students is $8$ and number of students to be selected or permutated is $6$. Find the number of ways of selection and permutation and state the difference between permutation and combination.

If total number of students is $15$ and number of students to be selected or permutated is $7$. Find the number of ways of selection and permutation and state the difference between permutation and combination.

If total number of students is $12$ and number of students to be selected or permutated is $4$. Find the number of ways of selection and permutation and state the difference between permutation and combination.

If total number of students is $8$ and number of students to be selected or permutated is $4$. Find the number of ways of selection and permutation and state the difference between permutation and combination.

How many different words may be made out of the letters of the word MATHEMATICS by taking $4$ letters at a time?

How many different permutations can be made with $3$ letters of FORECAST and $2$ letter of MILKY?

In how many different ways can $12$ different fruits be given away to $3$ boys, so that each boy gets at least one fruit?

A boat has a crew of $10$ men of which $3$ can row only on one side and $2$ only on the other. In how many ways can the crew be arranged in the boat?

There are $8$ prizes to be won in a lottery. A man picked up $5$ tickets from a bunch of $50$ tickets. In how many ways he can select the tickets, so that 
at least two prizes are guaranteed?

There are $8$ prizes to be won in a lottery. A man picked up $5$ tickets from a bunch of $50$ tickets. In how many ways he can select the tickets, so that 
exactly two prizes 
 

In how many different ways can $15$ distinct objects be divided

among $3$ persons equally?

In how many ways $5$ identical balls can be placed in $10$ identical boxes? [More than one ball cannot be placed in a box].

The results of $6$ cricket matches (win, loss or draw) are to be predicted. How may different forecasts can contain 
exactly $2$ correct results?

The results of $6$ cricket matches (win, loss or draw) are to be predicted. How may different forecasts can contain 
at least two correct results?