Permutations & Combinations
Words with Repeated-Letter Constraint
nta_pyq_2025_apr
Grade 11
Question:
The number of 6-letter words that can be formed using the letters of the word MATHS (M, A, T, H, S — all distinct), where every letter that appears in the word must appear at least twice, is
Step-by-Step Solution
Key Concept: Partition by the number of distinct letters used (1, 2, or 3), noting that using $k$ distinct letters in 6 positions with each appearing $\geq2$ times restricts compositions to $\{2,4\}$ or $\{3,3\}$ (for 2 letters) and $\{2,2,2\}$ (for 3 letters).
**1 distinct letter:** $\binom{5}{1}=5$ words.
**2 distinct letters** (distributions $(2,4)$, $(4,2)$, $(3,3)$):
$\binom{5}{2}\left(\dfrac{6!}{2!4!}+\dfrac{6!}{4!2!}+\dfrac{6!}{3!3!}\right)=10(15+15+20)=500$.
**3 distinct letters** (distribution $(2,2,2)$ only):
$\binom{5}{3}\cdot\dfrac{6!}{2!2!2!}=10\times90=900$.
Total $=5+500+900=1405$.
Correct Answer: 1405