Words of length $10$ are formed using the letters, $A, B, C, D, E, F, G, H, I, J$. Let $x$ be the number of such words where no letter is repeated ; and let $y$ be the number of such words where exactly one letter is repeated twice and no other letter is repeated. Then, $\frac{y}{9 x}=$
$5$
$4$
$8$
$9$
The total number of different combinations of one or more letters which can be made from the letters of the word ‘$MISSISSIPPI$’ is
If $^n{C_{r - 1}} = 36,{\;^n}{C_r} = 84$ and $^n{C_{r + 1}} = 126$, then the value of $r$ is
If $^{n + 1}{C_3} = 2{\,^n}{C_2},$ then $n =$
$\mathop \sum \limits_{0 \le i < j \le n} i\left( \begin{array}{l}
n\\
j
\end{array} \right)$ is equal to
The number of ways in which $21$ identical apples can be distributed among three children such that each child gets at least $2$ apples, is