$^n{P_r}{ \div ^n}{C_r}$ =

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

Find the number of words with or without meaning which can be made using all the letters of the word $AGAIN$. If these words are written as in a dictionary, what will be the $50^{\text {th }}$ word?

Let $S=\{1,2,3,5,7,10,11\}$. The number of nonempty subsets of $S$ that have the sum of all elements a multiple of $3$ , is $........$

How many words, with or without meaning, each of $2$ vowels and $3$ consonants can be formed from the letters of the word $\mathrm{DAUGHTER}$ ?

${ }^{n-1} C_r=\left(k^2-8\right){ }^n C_{r+1}$ if and only if: