The number of ways five alphabets can be chosen from the alphabets of the word $MATHEMATICS$,where the chosen alphabets are not necessarily distinct,is equal to :

If $\binom{18}{15} + 2\binom{18}{16} + \binom{17}{16} + 1 = \binom{n}{3}$,then find the value of $n$.

In a Mathematics examination,there are $20$ questions of equal marks. The question paper is divided into three sections: $A, B$,and $C$. $A$ student is required to attempt a total of $15$ questions,taking at least $4$ questions from each section. If section $A$ has $8$ questions,section $B$ has $6$ questions,and section $C$ has $6$ questions,then the total number of ways a student can select $15$ questions is:

$A$ class contains $b$ boys and $g$ girls. If the number of ways of selecting $3$ boys and $2$ girls from the class is $168$,then $b + 3g$ is equal to.

In a touring cricket team,there are $16$ players in all,including $5$ bowlers and $2$ wicket-keepers. How many teams of $11$ players can be chosen from these,such that the team includes exactly $3$ bowlers and $1$ wicket-keeper?

In how many ways can two balls of the same color be selected from $4$ distinct black balls and $3$ distinct white balls?