Question

The number of $4$ letter words that can be formed using the letters of the word $RAMANA$ is ?

Answer 1

$RAMANA$

The word has $1R,3A^{\prime }s,1M,1N$

let us solve by cases$:-$

$CaseI:$ all four letters are different i.e. $1R,1A,1M,1N$

Total words possible$=4!=24$

$CaseII:2$ distinct letters & $2A^{\prime }S$

Two distinct letters out of remaining $3$ letters (other than $A$) can be chosen in $3C_2$ ways

$\therefore$ Total words possible $=3C_2\frac{4!}{2!}=3\times \frac{24}{2}=36$

$CaseIII:1$ distinct letter 3A's$

One distinct letter can be chosen in $3C_1$ ways & therefore total words

$=3C_1\times \frac{4!}{3!}=3\times \frac{24}{6}=12$

Adding total possibilities$=24+36+12=72$