If $10 \cdot ^nC_2 = 3 \cdot ^{n+1}C_3$,then the value of $n$ is

The number of ways in which $17$ apples can be distributed among $4$ guests such that each guest gets at least $3$ apples is

There are $5$ types of ice creams available in a shop. $A$ child buys $6$ ice creams.
Statement-$1$: The child can buy $6$ ice creams in $^{10}C_5$ ways.
Statement-$2$: The number of ways the child can buy $6$ ice creams is equal to the number of ways to arrange $6$ '$A$'s and $4$ '$B$'s in a row.

The number of ordered triplets of positive integers which are solutions of the equation $x + y + z = 100$ is

If ${ }^{12} C_{2 k-1}={ }^{12} C_{k+1}$,then find $k$.

From a class of $25$ students,$10$ are to be chosen for an excursion party. There are $3$ students who decide that either all of them will join or none of them will join. In how many ways can the excursion party be chosen?