Question

How many different words can be formed with the letters of the word HARYANA? In how many of these $H$ and $N$ are together and how many of these begin with $H$ and end with $N$?

Answer 1

HARYANA. 7 letters
$3A^s,H,R,Y,N$.
(i) The number of words $=\frac{7!}{3!}=7\times 6\times 5\times 4=840$.
(ii) Treating $H$ and $N$ together we have
$7-2+1=6$ letters out of, which three are alike i.e., $A^s$ and hence they can be arranged in $\frac{6!}{3!}=120$ ways.
But $H$ and $N$ can be arranged amongst themselves in $2!=2$ ways.
(iii) Fix up $H$ in first and $N$ in last; we have 5 letters out of these are alike i.e. $A^s$ and hence the number of words is $\frac{5!}{3!}=5\times 4=20$.