Question

The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is ______.

Answer 1

1405

Answer 2

Let the available letters be $\{M, A, T, H, S\}$. Any letter used in the 6‑letter word must appear at least twice. Hence, the possible frequency distributions are:

1. Single letter only:

   A letter appears 6 times.

   • Ways: Choose 1 letter out of 5 $\rightarrow 5$ ways.

2. Two distinct letters:

   Possibilities:

   (a) (4,2) distribution:

   • Choose an ordered pair $(\text{letter for 4 times}, \text{letter for 2 times})$: $5 \times 4 = 20$ ways.
   • Arrangements: $\frac{6!}{4!2!} = 15$ ways.
   • Total: $20 \times 15 = 300$.

   (b) (3,3) distribution:

   • Choose 2 letters from 5: $\binom{5}{2} = 10$ ways.
   • Arrangements: $\frac{6!}{3!3!} = 20$ ways.
   • Total: $10 \times 20 = 200$.

   Combined for two letters: $300 + 200 = 500$.

3. Three distinct letters:

   Only possible if each appears twice, i.e., (2,2,2) distribution:

   • Choose 3 letters: $\binom{5}{3} = 10$ ways.
   • Arrangements: $\frac{6!}{2!2!2!} = 90$ ways.
   • Total: $10 \times 90 = 900$.

Total number of words $= 5 + 500 + 900 = 1405$.