Let $A = \left\{ {{a_1},\,{a_2},\,{a_3}.....} \right\}$ be a set containing $n$ elements. Two subsets $P$ and $Q$ of it is formed independently. The number of ways in which subsets can be formed such that $(P-Q)$ contains exactly $2$ elements, is

From $6$ different novels and $3$ different dictionaries, $4$ novels and $1$ dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. The number of such arrangements is :

$^n{C_r}\,{ \div ^n}{C_{r - 1}} = $

A set contains $2n + 1$ elements. The number of subsets of this set containing more than $n$ elements is equal to

A father with $8$ children takes them $3$ at a time to the Zoological gardens, as often as he can without taking the same $3$ children together more than once. The number of times he will go to the garden is

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

two are red cards and two are black cards,