In an election,the number of candidates is $1$ greater than the persons to be elected. If a voter can vote in $254$ ways,then the number of candidates is:

There are $15$ persons in a party and each person shakes hands with every other person exactly once. The total number of handshakes is:

Let $N$ be the set of positive integers. The number of distinct triplets $(x, y, z)$ satisfying $x, y, z \in N, x < y < z$ and $x+y+z=12$ is

Determine $n$ if $^{2n}C_{3} : ^{n}C_{3} = 11 : 1$.

In how many ways can $6$ persons be selected from $4$ officers and $8$ constables such that at least one officer is included?

Let $n$ and $k$ be positive integers such that $n \ge \frac{k(k + 1)}{2}$. The number of solutions $(x_1, x_2, ..., x_k)$ where $x_1 \ge 1, x_2 \ge 2, ..., x_k \ge k$ are all integers,satisfying $x_1 + x_2 + ... + x_k = n$,is