Let the number of elements in sets $A$ and $B$ be $5$ and $2$ respectively. Then the number of subsets of $A \times B$ each having at least $3$ and at most $6$ elements 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.

The English alphabet has $5$ vowels and $21$ consonants. How many words with two different vowels and two different consonants can be formed from the alphabet?

In how many ways can a committee of $6$ members be formed from $8$ men and $4$ women such that the committee contains at least $3$ women?

$A$ team of $11$ players is to be selected from $22$ players. If $2$ specific players must be included in every team and $4$ specific players must always be excluded,in how many ways can this selection be made?

$^nC_r + 2^nC_{r-1} + ^nC_{r-2} = $