Question

The number of permutations of the letters of the word ${}^{\prime }PROPORTIO{N}^{\prime }$, taken $4$ at a time, so that $3$ are alike and one is different is

• A. $15$
• B. $20$
• C. $25$
• D. $120$

Answer 1

The correct option is B $20$

Word PROPORTION has $P-2,R-2,O-3,T-1,I-1,N-1$
Now according to question we need to have 4 letter words of 3 alike and 1 different, Since PROPORTION has three O's and selection of 1 different from remaining is ${}^5{C}_1$
Now arrangements of 3 alike and 1 different is $\frac{4!}{3!}$
$\therefore$ required number of ways is ${}^5{C}_1\times \frac{4!}{3!}=20$