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 $...........$.
$412$
$411$
$413$
$414$
The number of words not starting and ending with vowels formed, using all the letters of the word $'UNIVERSITY'$ such that all vowels are in alphabetical order, is
The number of matrices of order $3 \times 3$, whose entries are either $0$ or $1$ and the sum of all the entries is a prime number, is$....$
If $^{{n^2} - n}{C_2}{ = ^{{n^2} - n}}{C_{10}}$, then $n = $
If $^{15}{C_{3r}}{ = ^{15}}{C_{r + 3}}$, then the value of $r$ is