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

Statement-$1:$ The number of ways of distributing $10$ identical balls in $4$ distinct boxes such that no box is empty is $^9C_3$.
Statement-$2:$ The number of ways of choosing any $3$ places from $9$ different places is $^9C_3$.

The number of ways in which $10$ persons can go in two boats so that there may be $5$ on each boat,supposing that two particular persons will not go in the same boat is

Find the number of ways of selecting $4$ pens and $3$ pencils from a packet of $8$ pens and $5$ pencils.

If $\binom{n}{3} + \binom{n}{4} > \binom{n+1}{3}$,then:

There are $15$ players in a cricket team,out of which $6$ are bowlers,$7$ are batsmen,and $2$ are wicketkeepers. The number of ways a team of $11$ players can be selected from them so as to include at least $4$ bowlers,$5$ batsmen,and $1$ wicketkeeper is $.....$