The number of ways in which $21$ identical apples can be distributed among three children such that each child gets at least $2$ apples,is

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

There are $10$ bulbs in a room. Each of them can be switched on independently. In how many ways can the room be illuminated?

What is the number of ways of choosing $4$ cards from a pack of $52$ playing cards? In how many of these are two red cards and two black cards?

$A$ group of $9$ students,$s_1, s_2, \ldots, s_9$,is to be divided into three teams $X, Y$,and $Z$ of sizes $2, 3$,and $4$,respectively. Suppose that $s_1$ cannot be selected for team $X$,and $s_2$ cannot be selected for team $Y$. The number of ways to form such teams is:

There are two urns. Urn $A$ contains $3$ distinct red balls,and Urn $B$ contains $9$ blue balls. If two balls are selected at random from each urn and swapped,in how many ways can this operation be performed?