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

$\mathrm{In} \mathrm{the} \mathrm{word} \mathrm{SARANAM}$

$\mathrm{There} \mathrm{are} \mathrm{AAA},\mathrm{S},\mathrm{R},\mathrm{N},\mathrm{M}$

$$\begin{gathered} \mathrm{Number} \mathrm{of} 5 \mathrm{letter} \mathrm{words} \\ \mathrm{with} \mathrm{all} \mathrm{different} \mathrm{letters}\left\{\mathrm{S},\mathrm{A},\mathrm{R},\mathrm{N},\mathrm{M}\right\}=5! \end{gathered}$$

$$\begin{gathered} \mathrm{Number} \mathrm{of} 5 \mathrm{letter} \mathrm{words} \mathrm{with} \\ \left\{\mathrm{A},\mathrm{A}\right\} \mathrm{and} 3 \mathrm{letters} \mathrm{from} \left\{\mathrm{S},\mathrm{R},\mathrm{N},\mathrm{M}\right\}={ }^4{\mathrm{C}}_3\times \frac{5!}{2!}=4\times 60=240 \end{gathered}$$

$$\begin{gathered} \mathrm{Number} \mathrm{of} 5 \mathrm{letter} \mathrm{words} \mathrm{with} \\ \mathrm{AAA} \mathrm{and} \mathrm{two} \mathrm{letters} \mathrm{from}\left\{\mathrm{S},\mathrm{R},\mathrm{N},\mathrm{M}\right\}={ }^4{\mathrm{C}}_2\times \frac{5!}{3!}=6\times 20=120 \end{gathered}$$

$\mathrm{Total} \mathrm{number} \mathrm{of} 5 \mathrm{letter} \mathrm{words}=120+240+120=480$