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 8C4 × 4! = 1680
 
ii) Three pairs alike letters MM, AA, TT. One pair can be chosen by 3C1 = 3 ways 
Remaining two letters can be chosen by 7C2 
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 × 7C2 × 4!/2! = 756
 
iii) Three pairs of 2 alike letters. Two pairs can be chosen by 3C2
 3C2 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 = 3C× 4!/2! × 1/2! = 18
 
Total number of ways = 1680 + 756 + 18 = 2454

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