Let $A_1,A_2,……..A_{11}$ are players in a team with their T-shirts numbered $1,2,…..11$. Hundred gold coins were won by the team in the final match of the series. These coins is to be distributed among the players such that each player gets atleast one coin more than the number on his T-shirt but captain and vice captain get atleast $5$ and $3$ coins respectively more than the number on their respective T-shirts, then in how many different ways these coins can be distributed ?

In how many ways can a committee be formed of $5$ members from $6$ men and $4$ women if the committee has at least one woman

The number of ways in which four letters of the word $‘MATHEMATICS$’ can be arranged is given by

Let $S=\{1,2,3,5,7,10,11\}$. The number of nonempty subsets of $S$ that have the sum of all elements a multiple of $3$ , is $……..$

An urn contains $5$ red marbles, $4$ black marbles and $3$ white marbles. Then the number of ways in which $4$ marbles can be drawn so that at the most three of them are red is