$^{2n}C_4 : ^nC_3 = 99 : 4 \Rightarrow n = $

$A$ committee of $5$ is to be formed out of $6$ men and $4$ ladies. The number of ways this can be done,when at most $2$ ladies are included,is

Let $S_r = \{(x, y, z) : x + y + z = 11, x \geq r, y \geq r, z \geq r, x, y, z, r \in \mathbb{Z}\}$ and $n(S_r)$ represents the number of elements in $S_r$. Then $n(S_2) + n(S_3) + n(S_4) = $

The least value of $n$ for which ${ }^{(n-1)} C_2 + { }^{(n-1)} C_3 > { }^n C_2$ is

The number of ways in which $6$ distinct things can be distributed into $2$ boxes so that no box is empty is

How many distinct combinations can be formed using one or more letters from the word $MISSISSIPPI$?