Consider a group of $5$ boys and $7$ girls. The number of different teams,consisting of $2$ boys and $3$ girls that can be formed from this group if there are two specific girls $A$ and $B$,who refuse to be the members of the same team,is

In how many ways can a team of $10$ players be selected from $22$ players if $6$ particular players are always to be included and $4$ particular players are always excluded?

If $^{n}C_{8} = ^{n}C_{2}$,find $^{n}C_{2}$.

The English alphabet has $5$ vowels and $21$ consonants. How many words with two different vowels and two different consonants can be formed from the alphabet?

$A$ committee of $7$ members is to be formed from $6$ boys and $4$ girls such that the number of boys is greater than the number of girls. In how many ways can this be done?

$A$ question paper has two sections $A$ and $B$,in which section-$A$ has $8$ questions and section-$B$ has $6$ questions. $A$ student has to answer a total of $10$ questions,choosing at least $4$ questions from section-$A$ and at least $3$ questions from section-$B$. The number of ways a student can answer the paper is: