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:
Solution:
We have 5 distinct letters: {M, A, T, H, S}.
We need to form a 6-letter word where every chosen letter appears at least 2 times.
Let $k$ be the number of distinct letters used.
Case 1: Use only 1 distinct letter ($k=1$)
Select 1 letter from 5: ${}^{5}C_{1} = 5$.
The letter must repeat 6 times (e.g., MMMMMM).
Number of ways = $5 \times \frac{6!}{6!} = 5$.
Case 2: Use 2 distinct letters ($k=2$)
Select 2 letters from 5: ${}^{5}C_{2} = 10$.
We need to distribute 6 positions between these 2 letters such that each appears $\ge 2$ times.
Possible distributions of counts $(n_1, n_2)$ where $n_1 + n_2 = 6$:
1. $(4, 2)$: Arrangements = $\frac{6!}{4!2!} = 15$. (Can be $A^4 B^2$ or $A^2 B^4$, so $\times 2$).
2. $(3, 3)$: Arrangements = $\frac{6!}{3!3!} = 20$.
Total arrangements for 2 letters = $10 \times [(15 \times 2) + 20] = 10 \times [30 + 20] = 500$.
Case 3: Use 3 distinct letters ($k=3$)
Select 3 letters from 5: ${}^{5}C_{3} = 10$.
Possible distributions where sum is 6 and each $\ge 2$:
Only one possibility: $(2, 2, 2)$.
Arrangements = $\frac{6!}{2!2!2!} = \frac{720}{8} = 90$.
Total arrangements for 3 letters = $10 \times 90 = 900$.
Total Words
Sum of all cases = $5 + 500 + 900 = 1405$.
Ans. (1405)