The value of ${ }^{34}C_5 + \sum_{i=0}^4 {}^{38-i}C_4$ is

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

If $(2 \leq r \leq n)$,then ${}^{n}C_{r} + 2 \cdot {}^{n}C_{r+1} + {}^{n}C_{r+2}$ is equal to:

The number of ways in which an examiner can assign $30$ marks to $8$ questions,awarding not less than $2$ marks to any question is

From $5$ consonants and $5$ vowels,how many words can be formed using $3$ consonants and $2$ vowels?

In how many ways can a committee consisting of one or more members be formed out of $12$ members of the Municipal Corporation?