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Q.
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
TS EAMCET 2019
Solution:
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
$={ }^{7} P_{4}+\frac{{ }^{3} C_{1} \times{ }^{6} C_{2} \times 4 !}{2 !}+\frac{{ }^{3} C_{2} \times 4 !}{2 ! 2 !}$
$=7 \times 6 \times 5 \times 4+\frac{3 \times 6 \times 5 \times 4 \times 3 \times 2 !}{2 ! 2 !}+\frac{3 \times 4 \times 3 \times 2 !}{2 ! 2 !}$
$=840+540+18=1398$