There are $25$ candidates for $12$ positions,out of which $5$ are from the reserved category. If $3$ positions are reserved and the remaining are open,in how many ways can the selection be made?

The number of ways in which $35$ apples can be distributed among $3$ boys so that each can have any number of apples,is

$^{n-1}C_3 + ^{n-1}C_4 > ^nC_3$,then the value of $n$ is

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

If $4$ red balls and $5$ green balls are selected from $n$ balls,and the sum of both selections is greater than ${}^{n+1}C_4$,then the value of $n$ is equal to:

The number of values of $n \in N$ for which ${ }^{n+2} C_2 : { }^{n+3} C_1 = 4 : 2$ is