How many 4-letter words can be formed from MATHEMATICS

Question

How many 4-letter words (with or without meaning) can be formed using the letters of the word MATHEMATICS?

(JEE Main 2023, similar pattern)

Solution — Step by Step

MATHEMATICS has 11 letters: M(2), A(2), T(2), H(1), E(1), I(1), C(1), S(1).

So we have 8 distinct letters, with M, A, T each appearing twice. We need to form 4-letter arrangements, considering these repetitions.

Case 1: All 4 letters distinct. Choose 4 from the 8 distinct letters and arrange: $\binom{8}{4} \times 4! = 70 \times 24 = 1680$

Case 2: Exactly one pair of identical letters + 2 distinct letters. Choose which letter repeats (M, A, or T — 3 choices). Choose 2 more from the remaining 7 distinct letters: $\binom{7}{2} = 21$ . Arrange these 4 letters (with 2 identical): $\frac{4!}{2!} = 12$ .

Total: $3 \times 21 \times 12 = 756$

Case 3: Two pairs of identical letters. Choose 2 letters from {M, A, T} to form two pairs: $\binom{3}{2} = 3$ . Arrange 4 letters (2 pairs): $\frac{4!}{2! \cdot 2!} = 6$ .

Total: $3 \times 6 = 18$

\text{Total} = 1680 + 756 + 18 = \mathbf{2454}

Why This Works

When letters repeat, we can’t use a simple $^nP_r$ formula. The case-by-case approach based on the repetition pattern is the standard technique. Each case has a distinct structure:

All different: straightforward permutation
One pair + two singles: divide by $2!$ to avoid overcounting the identical pair
Two pairs: divide by $2! \times 2!$

This exhausts all possible patterns for 4 letters drawn from our set, since no letter appears more than twice (ruling out cases like three identical letters).

Alternative Method — Generating function (for verification)

The generating function for each letter type is:

Letters with frequency 2 (M, A, T): $(1 + x + x^2/2!)$ each
Letters with frequency 1 (H, E, I, C, S): $(1 + x)$ each

The number of 4-letter arrangements is $4!$ times the coefficient of $x^4$ in the product of all these generating functions. This confirms 2454 but is slower to compute by hand.

This exact question — “MATHEMATICS” — is one of the most-repeated PYQ patterns in JEE. The answer 2454 is worth memorising. But more importantly, learn the case-splitting technique — it applies to any word with repeated letters (MISSISSIPPI, COMMITTEE, etc.).

Common Mistake

Students often miss Case 3 entirely. They handle “all distinct” and “one pair + two distinct” but forget that two pairs is also possible (e.g., MMAA, MMTT, AATT and their arrangements). Missing this case gives 2436 instead of 2454. Always enumerate all possible repetition patterns before calculating.

Question

Solution — Step by Step

Why This Works

Alternative Method — Generating function (for verification)

Common Mistake

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