Out of $10$ white,$9$ black and $7$ red balls,the number of ways in which selection of one or more balls can be made,is

If $\frac{n!}{2!(n-2)!}$ and $\frac{n!}{4!(n-4)!}$ are in the ratio $2:1$,find the value of $n$.

The number of ways in which $35$ apples can be distributed among $3$ boys so that each can have any number of apples,is

There are $6$ members to be selected for a group to be sent abroad from a total of $12$ members. In how many ways can the selection be made such that a particular member is always excluded?

If all the letters of the word $'GANGARAM'$ are arranged,then the number of words in which exactly two vowels are together but no two $'G'$ occur together is-

$A$ person is permitted to select at least one and at most $n$ coins from a collection of $(2n + 1)$ distinct coins. If the total number of ways in which he can select coins is $255$,then $n$ equals