Question

The number of 4-letter words, with or without meaning, which can be formed using the letters PQRPRSTUVP, is :

Hint:

When dealing with repeated letters, always list the counts first. It prevents missing cases like "2 alike + 2 alike" which are common pitfalls.

Answer 1

Correct Answer: 1422

Step 1: Understanding the Concept:
Identify the frequency of letters in the given set: P (3), R (2), Q (1), S (1), T (1), U (1), V (1). Total 10 letters, with 7 distinct letters.
Step 2: Key Formula or Approach:
Categorize the cases for a 4-letter selection: 1. 3 alike, 1 different. 2. 2 alike, 2 alike. 3. 2 alike, 2 different. 4. All 4 different.
Step 3: Detailed Explanation:
- Case 1 (3 alike, 1 different): Choose 1 letter from {P} (1 way) and 1 from remaining 6 distinct letters (\(^6C_1\)). Arrangements: \( 1 \times 6 \times \frac{4!}{3!} = 24 \). - Case 2 (2 alike, 2 alike): Choose 2 letters from {P, R}. Arrangements: \( ^2C_2 \times \frac{4!}{2!2!} = 6 \). - Case 3 (2 alike, 2 different): Choose 1 from {P, R} for pair (\(^2C_1\)) and 2 from remaining 6 distinct letters (\(^6C_2\)). Arrangements: \( 2 \times 15 \times \frac{4!}{2!} = 360 \). - Case 4 (All different): Choose 4 from 7 distinct letters (\(^7C_4\)). Arrangements: \( 35 \times 4! = 840 \).
Total = \( 24 + 6 + 360 + 840 = 1230 \).
Step 4: Final Answer:
The total number of words is 1230.