If ${ }^{2 n} C_{r} = { }^{2 n} C_{r+2}$,find $r$.

$^{47}C_4 + \sum_{r=1}^{5} {}^{52-r}C_3 = $

How many $4$-digit numbers,divisible by $5$,can be formed using the digits $0, 3, 5, 7,$ and $9$ if repetition of digits is not allowed?

$A$ father with $8$ children takes them $3$ at a time to the Zoological gardens,as often as he can without taking the same $3$ children together more than once. The number of times each child will go to the garden is

$^{14}C_4 + \sum_{j=1}^4 {^{18-j}C_3}$ is equal to

If $^{n^2 - n}C_2 = ^{n^2 - n}C_{10}$,then $n = $