Question

Number of 4-letter words (with or without meaning) formed from the letters of the word \( \text{PQRSSSTTUVW} \) is:

• A. \(1232\)
• B. \(1400\)
• C. \(1422\)
• D. \(1162\)

Hint:

When letters repeat, count cases separately based on how many times repeated letters are used, and apply division by factorials carefully.

Answer 1

The Correct Option is C

Alternative Method (Repetition-Based Case Analysis):

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The given word contains 11 letters, with repetitions:

• S appears 3 times
• T appears 2 times
• P, Q, R, U, V, W appear once each

We form 4-letter meaningful arrangements without exceeding available repetitions. Since order matters, permutations are used.

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Case I: All letters different

Choose any 4 distinct letters from the 8 distinct letters:

{P, Q, R, S, T, U, V, W}

Number of ways:

⁸P₄ = 8 × 7 × 6 × 5 = 1680

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Case II: Exactly one repeated letter

(a) One pair of S

After fixing S twice, choose 2 more distinct letters from the remaining 7 letters:

C(7, 2)

Number of arrangements of 4 letters with one pair:

4! / 2! = 12

Total words:

C(7, 2) × 12 = 252

(b) One pair of T

Same calculation applies:

252

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Case III: Two different pairs (S and T)

Letters involved: S, S, T, T

Number of arrangements:

4! / (2! × 2!) = 6

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Case IV: Three identical letters

Only S can appear three times. Choose 1 more letter from the remaining 7 letters:

C(7, 1)

Number of arrangements:

4! / 3! = 4

Total words:

7 × 4 = 28

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Final Calculation:

1680 + 252 + 252 + 6 + 28 = 1422

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Final Answer:

Number of different 4-letter words = 1422