Find the number of all possible arrangement of the letters of the word

1.	Find the number of all possible arrangement of the letters of the word “MATHEMATICS” taken four from at a time.
Answer» The word MATHEMATICS consist of 11 letters: (M,M), (A,A), (T,T), H,E,I,C,S Case 1: In this case 2 similar and 2 similar letters are selected, number of arrangements = ³C₂ × \(\frac{4!}{2!\space2!}=756\) Case 2: In this case all 4 letters selected are different, number of arrangements = ⁸C₄ × 4! = 1680 Therefore, total number of arrangement = 18 + 756 + 1680 = 2454

Find the number of all possible arrangement of the letters of the word “MATHEMATICS” taken four from at a time.

Answer»

The word MATHEMATICS consist of 11 letters:

(M,M), (A,A), (T,T), H,E,I,C,S

Case 1: In this case 2 similar and 2 similar letters are selected, number of arrangements

= ³C₂ × \(\frac{4!}{2!\space2!}=756\)

Case 2: In this case all 4 letters selected are different, number of arrangements = ⁸C₄ × 4!

= 1680

Therefore, total number of arrangement = 18 + 756 + 1680

= 2454