Question

The total number of $4-$letter words that can be made by using the letters of word $TOMATO$ is

Answer 1

Given word : $TOMATO$
Total letters $=6(2T^{\prime }s+2O^{\prime }s+1M^{\prime }s+1A^{\prime }s)$
Total different letters $=4$

Case $\left(1\right):$ When all $4$ letters are different
Number of ways $={}^4{C}_4\cdot 4!=24$

Case $\left(2\right):$ When $2$ letters are alike (either $2T^{\prime }s\text{or}2O^{\prime }s$) and $2$ different letter
Number of ways $={}^2{C}_1\cdot {}^3{C}_2\cdot \frac{4!}{2!}=72$

Case $\left(3\right):$ When $2T^{\prime }s$ and $2O^{\prime }s$ are taken
Number of ways $={}^2{C}_2\cdot \frac{4!}{2!\cdot 2!}=6$
$\therefore$ Total required words $=24+72+6=102$