Evaluate $\frac{n!}{r!(n-r)!}$ when $n=5$ and $r=2$.

The number of ways in which $15$ identical gold coins can be distributed among $3$ persons such that each one gets at least $3$ gold coins is

If $n$ objects are arranged in a row,then the number of ways of selecting three of these objects such that no two of them are next to each other is:

The total number of ways of selecting $4$ letters from all the letters of the word $TSEAMCET$ is

$A$ student is allowed to select at most $n$ books from a collection of $(2n+1)$ books. If the total number of ways in which one can select at least one book is $255$,then the value of $n$ is

The difference between the maximum value of ${}^6C_r$ and ${}^nC_3$ is $16$. Then $n=$