Question

There are eight persons – P, Q, R, S, T, U, V and W, standing in a row and four distinct articles A, B, C and D are to be given to four people. No four neighboring persons receive an article. How many ways can this distribution be done?

• A. 1680 ways
• B. 1560 ways
• C. 1440 ways
• D. 1380 ways
• E. 1320 ways

Answer 1

The Correct Option is B

Step 1: Understand the problem.
There are eight persons – P, Q, R, S, T, U, V, and W standing in a row, and four distinct articles A, B, C, and D are to be given to four people. The constraint is that no four neighboring persons can receive an article. We need to find how many ways this distribution can be done.

Step 2: Identify the constraints and the solution approach.
- There are eight persons, and four articles are to be given to four distinct persons. - No four neighboring persons can receive an article. This means that the four selected persons must not be consecutive.

Step 3: Select four persons from the eight.
First, we need to select four persons from the eight available. The number of ways to select four persons from eight is given by the combination formula \( \binom{n}{k} \), where \( n \) is the total number of persons, and \( k \) is the number of persons to be selected: \[ \binom{8}{4} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70 \] So, there are 70 ways to choose four persons from the eight.

Step 4: Assign the four distinct articles to the selected persons.
After selecting four persons, we need to assign four distinct articles (A, B, C, D) to these selected persons. The number of ways to assign four distinct articles to four persons is the number of permutations of the four articles: \[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]

Step 5: Combine the results.
To find the total number of ways to distribute the four articles, we multiply the number of ways to select the four persons by the number of ways to assign the articles: \[ 70 \times 24 = 1680 \]

Step 6: Adjust for the constraint.
The constraint is that no four persons can be consecutive. So, we subtract the number of ways in which all four selected persons are consecutive. If four persons are consecutive, they can only be chosen in one of five possible ways (the first set of four persons, the second, and so on). For each of these 5 sets, the articles can be assigned in \( 4! = 24 \) ways. Therefore, the number of ways in which all four persons are consecutive is: \[ 5 \times 24 = 120 \] Subtracting this from the total number of ways: \[ 1680 - 120 = 1560 \]

Step 7: Conclusion.
The total number of ways to distribute the articles, satisfying the condition, is 1560.

Final Answer:
The correct answer is (B): 1560 ways.