JEE Main 2025 Jan 29 Shift 1, Mathematics Q24: Combinations

JEE Main2025Jan 29, Shift 1Mathematics

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 ______.

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Solution

Each letter used must appear $\geq 2$ times in the 6-letter word. With 5 distinct letters ${M, A, T, H, S}$ , possible multiplicity profiles summing to $6$ are:

One distinct letter (profile 6): choose 1 of 5 letters, fill all 6 positions. Count $= 5.$

Two distinct letters: profiles are $(4, 2)$ and $(3, 3) .$

$(4, 2)$ : choose the 2 letters $(2 5) = 10$ , decide which has multiplicity 4 (factor of 2), then arrange: $10 \cdot 2 \cdot \frac{6 !}{4 ! 2 !} = 10 \cdot 2 \cdot 15 = 300.$

$(3, 3)$ : choose the 2 letters $(2 5) = 10$ , arrange: $10 \cdot \frac{6 !}{3 ! 3 !} = 10 \cdot 20 = 200.$

Subtotal for two letters $= 300 + 200 = 500.$

Three distinct letters (profile $2, 2, 2$ ): choose 3 of 5 letters $(3 5) = 10$ , arrange: $10 \cdot \frac{6 !}{2 ! 2 ! 2 !} = 10 \cdot 90 = 900.$

Higher counts of distinct letters are impossible (4 distinct letters each $\geq 2$ needs $\geq 8$ slots).

Total $= 5 + 500 + 900 = 1405.$

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JEE Main 2025 Jan 29 Shift 1, Mathematics Q24: Combinations

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