The number of groups that can be made from $5$ different green balls,$4$ different blue balls,and $3$ different red balls,if at least $1$ green and $1$ blue ball is to be included.

If $C(n, 8) = C(n, 6)$,find $C(n, 2)$.

If the number of $5$-element subsets of the set $A = \{a_1, a_2, \dots, a_{20}\}$ of $20$ distinct elements is $k$ times the number of $5$-element subsets containing $a_4$,then $k$ is

The set $S = \{1, 2, 3, \ldots, 12\}$ is to be partitioned into three sets $A, B, C$ of equal size. Thus $A \cup B \cup C = S$ and $A \cap B = B \cap C = C \cap A = \emptyset$. The number of ways to partition $S$ is

The greatest possible number of points of intersection of $8$ straight lines and $4$ circles is

Let $A_1, A_2, ......, A_{11}$ be players in a team with their $T$-shirts numbered $1, 2, ......, 11$. One hundred gold coins were won by the team. These coins are to be distributed among the players such that each player $A_i$ gets at least $i$ coins,plus an additional coin (i.e.,at least $i+1$ coins). Additionally,the captain (assume $A_1$) and vice-captain (assume $A_2$) must receive at least $5$ and $3$ coins more than their $T$-shirt numbers respectively. In how many different ways can these coins be distributed?