How many words can be formed by taking 4 letters at a time out of the letters of the words MATHEMATICS?

### Asked by Kashifimam866 | 20th Feb, 2019, 05:58: AM

Expert Answer:

### The word contain 11 letters with repetition of letters.
We can choose 4 letters by
i) all the four letters are distinct
ii) Two distinct and two alike
iii) Two alike of one kind and two alike of other kind
i) 8 different letters are M, A, T, H, E, I, C, S
This can be chosen by ^{8}C_{4} × 4! = 1680
ii) Three pairs alike letters MM, AA, TT. One pair can be chosen by ^{3}C_{1} = 3 ways
Remaining two letters can be chosen by ^{7}C_{2}
Each such groups has 4 letters out of which 2 are alike.
Hence, it can be chosen by 4!/2!
Total number of ways = 3 × ^{7}C_{2} × 4!/2! = 756
iii) Three pairs of 2 alike letters. Two pairs can be chosen by ^{3}C_{2}
^{3}C_{2 }groups of 4 letters each.
There are 4 letters of which 2 alike of one kind and two alike of other kind in each group.
That can be arranged by 4!/2! × 1/2!
Total number of ways = ^{3}C_{2 }×_{ }4!/2! × 1/2! = 18
Total number of ways = 1680 + 756 + 18 = 2454

^{8}C

_{4}× 4! = 1680

^{3}C

_{1}= 3 ways

^{7}C

_{2}

^{7}C

_{2}× 4!/2! = 756

^{3}C

_{2}

^{3}C

_{2 }groups of 4 letters each.

^{3}C

_{2 }×

_{ }4!/2! × 1/2! = 18

### Answered by Sneha shidid | 20th Feb, 2019, 10:05: AM

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