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 _______.

We form 6-letter words from letters of MATHS (5 distinct letters), with condition:

Any letter used must appear at least twice.

Possible repetition patterns for total 6 letters:

Case 1: (2+2+2)

Choose 3 letters from 5:
$$\binom{5}{3}=10$$

Arrange:
$$\frac{6!}{2!2!2!}=90$$

Total:
$$10\times90=900$$

Case 2: (4+2)

Choose letter repeated 4 times:
5

Choose letter repeated 2 times from remaining 4:
4

Arrange:
$$\frac{6!}{4!2!}=15$$

Total:
$$5\times4\times15=300$$

Case 3: (3+3)

Choose 2 letters from 5:
$$\binom{5}{2}=10$$

Arrange:
$$\frac{6!}{3!3!}=20$$

Total:
$$10\times20=200$$

Case 4: (2+2+1+1) ❌ not allowed

(two letters appear once)

Case 5: (6)

All same letter:

5

Total 900+300+200+5=1405