Question

The number of different permutations of letters that can be formed by taking $4$ letters at a time from the letters of the word ' $REPETITION$ ' is

• A. 210
• B. 720
• C. 1398
• D. 5040

Answer 1

Answer: (C)

In given word $REPETITION$, the letters $R,P,O$ and $N$ come one time while the letters $E,T$ and $I$ come two times.
Now, the number of different permutations of letters that can be formed by taking 4 letters at a time can be made in following ways
(i) All are different $={}^7{P}_4$
(ii) $2$ are different and one pair $=\frac{{}^3{C}_1\times {}^6{C}_2\times 4!}{2!}$
(iii) $2$ pairs $=\frac{{}^3{C}_2\times 4!}{2!\times 2!}$
So, required number of permutations
$={}^7{P}_4+\frac{{}^3{C}_1\times {}^6{C}_2\times 4!}{2!}+\frac{{}^3{C}_2\times 4!}{2!2!}$
$=7\times 6\times 5\times 4+\frac{3\times 6\times 5\times 4\times 3\times 2!}{2!2!}+\frac{3\times 4\times 3\times 2!}{2!2!}$
$=840+540+18=1398$