Step 1: The word "MATHS" contains 5 distinct letters: M, A, T, H, and S. We are required to form 6-letter words where each letter that appears must appear at least twice.
Step 2: The only way to satisfy the condition of having each letter that appears at least twice in a 6-letter word is by using exactly 2 of each of 2 letters.
Step 3: The number of ways to choose 2 letters from the 5 available letters is \( \binom{5}{2} \), and for each choice of letters, the 6 positions can be arranged in \( \frac{6!}{2!2!} \) ways. Thus, the total number of such 6-letter words is calculated.
Let $ P_n = \alpha^n + \beta^n $, $ n \in \mathbb{N} $. If $ P_{10} = 123,\ P_9 = 76,\ P_8 = 47 $ and $ P_1 = 1 $, then the quadratic equation having roots $ \alpha $ and $ \frac{1}{\beta} $ is: