The number of 5-letter words that can be formed by using the letters of the word SARANAM is

There are 7 letters in the word 'SARANAM' of which there are $3 \mathrm{~A}$ and all other are distinct. The five letter words may consist of (i) all different letters (using S, A, R, N, M) $\therefore$ Required words $$ =5 !=120 $$ (ii) 2 alike and other different (using $2 \mathrm{~A}$ and 3 other) $\therefore$ Required words $$ ={ }^4 C_3 \times \frac{5 !}{2 !}=4 \times \frac{120}{2}=240 $$ (iii) 3 alike and other different (using 3A and 2 other) $\therefore$ Required words $={ }^4 C_2 \times \frac{5 !}{3 !}=\frac{4 \times 3}{2 \times 1} \times \frac{120}{6}=120$ $\therefore$ Total words $=120+240+120=480$