Question

Find the number of all possible arrangements of the letters of the word MATHEMATICS taken 4 at a time ?

Answer 1

There are 8 distinct letters: M-A-T-H-E-I-C-S. 3 letters M, A, and T are represented twice (double letter). Selected 4 letters can have following 3 patterns: 

1. abcd - all 4 letters are different: 8P4=1680 (choosing 4 distinct letters out of 8, when order matters) or 8C4∗4!=1680 (choosing 4 distinct letters out of 8 when order does not matter and multiplying by 4! to get different arrangement of these 4 distinct letters

2. aabb - from 4 letters 2 are the same and other 2 are also the same: 3C2∗4!2!2!=18 - 3C2 choosing which two double letter will provide two letters (out of 3 double letter - MAT), multiplying by 4!2!2! to get different arrangements (for example MMAA can be arranged in 4!2!2! # of ways.

3. aabc - from 4 letters 2 are the same and other 2 are different: 3C1∗7C2∗4!2!=756 - 3C1 choosing which letter will proved with 2 letters (out of 3 double letter - MAT), 7C2 choosing third and fourth letters out of 7 distinct letters left and multiplying by 4!2! to get different arrangements (for example MMIC can be arranged in 4!2! # of ways

1680+18+756=2454


hope it helped.....☺️

Answer 2

Answer:

Case 1:2 similar and 2 similar letters

=C(3,2) ×4!÷2!.2!

=18

Case2:2 similar and 2 different letters

=C(3,1)×C(7,2) ×4!÷2!

=756

Case 3:all different letters

=C(8,4)×4!

=1680

therefore, total arrangements =18+756+1680

=2454