$A$ bag contains $5$ black and $6$ red balls. Determine the number of ways in which $2$ black and $3$ red balls can be selected.

The number of integers $x, y, z, w$ satisfying $x+y+z+w=25$ and $x, y, z \geq -1, w \geq 1$ is

The number of $4$-digit numbers without repetition that can be formed using the digits $1, 2, 3, 4, 5, 6, 7$ in which each number has two odd digits and two even digits is:

If $^nC_{r-2} = 36$,$^nC_{r-1} = 84$,and $^nC_r = 126$,then the value of $^nC_{2r}$ is

Out of $7$ consonants and $4$ vowels,the number of words consisting of $3$ consonants and $2$ vowels that can be formed is:

The number of ways of selecting a committee of $30$ persons from $20$ boys,$20$ girls,and $20$ teachers such that the participation of the number of boys,girls,and teachers in that committee is equal,is