Question

The number of selections of $6$ different letters that can be made from the words $\text{NISHIT}$ and $\text{RAHUL}$ so that each selection consists of $3$ letters from each word, is

Answer 1

Here $H$ is the common letter in words $\text{NISHIT}$ and $\text{RAHUL}$
Now, for selecting six different letters, we must select $H$ either from word $\text{RAHUL}$ or from word $\text{NISHIT}$.

Hence, for the possible selections, we have
Case $1:$ $H$ included in $\text{RAHUL}$ (excluded in$\text{NISHIT}$)
Number of selections $={}^4{C}_2\cdot {}^4{C}_3$

Case $2:$ $H$ excluded in $\text{RAHUL}$ (no constraint in $\text{NISHIT}$)
Number of selections $={}^4{C}_3\cdot {}^5{C}_3$

Total count $={}^4{C}_2\cdot {}^4{C}_3+{}^4{C}_3\cdot {}^5{C}_3$
$=24+40=64$