$\sum\limits_{r = 0}^m {^{n + r}{C_n} = } $
$^{n + m + 1}{C_{n + 1}}$
$^{n + m + 2}{C_n}$
$^{n + m + 3}{C_{n - 1}}$
None of these
The number of seven digit positive integers formed using the digits $1,2,3$ and $4$ only and sum of the digits equal to $12$ is $...........$.
The number of onto functions $f$ from $\{1, 2, 3, …, 20\}$ only $\{1, 2, 3, …, 20\}$ such that $f(k)$ is a multiple of $3$, whenever $k$ is a multiple of $4$, is
How many words, with or without meaning, can be formed using all the letters of the word $\mathrm{EQUATION}$ at a time so that the vowels and consonants occur together?
In how many ways $5$ speakers $S_1,S_2,S_3,S_4$ and $S_5$ can give speeches one after the other if $S_3$ wants to speak after $S_1$ & $S_2$
If $^{n}{P_4} = 24.{\,^n}{C_5},$ then the value of $n$ is