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

Concept:

Total letters in the word: \(11\)
Letter frequencies: \[ \text{S appears 3 times},\quad \text{T appears 2 times} \] \[ \text{P, Q, R, U, V, W appear once each} \]
Words are formed without exceeding the available repetitions.
Order of letters matters.
Step 1: Count words with all distinct letters. Distinct letters available: \[ \{P, Q, R, S, T, U, V, W\} \Rightarrow 8 \text{ letters} \] \[ \text{Number} = {}^8P_4 = 8 \times 7 \times 6 \times 5 = 1680 \]
Step 2: Words with one pair. (a) Pair of S: Choose 2 other distinct letters from remaining 7: \[ \binom{7}{2} \] Arrangements: \[ \frac{4!}{2!} = 12 \] \[ \text{Total} = \binom{7}{2} \times 12 = 252 \] (b) Pair of T: Similarly: \[ \binom{7}{2} \times 12 = 252 \]
Step 3: Words with two pairs (S and T). \[ \frac{4!}{2!2!} = 6 \]
Step 4: Words with three same letters (only SSS possible). Choose 1 more letter from remaining 7: \[ \binom{7}{1} \] Arrangements: \[ \frac{4!}{3!} = 4 \] \[ \text{Total} = 7 \times 4 = 28 \]
Step 5: Add all cases. \[ 1680 + 252 + 252 + 6 + 28 = 1422 \]