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 neighbouring persons receive an article. How many ways can this distribution be done?

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

Hint:

Count the ways to choose 4 positions out of 8 in a row, remove the cases where all four chosen positions are consecutive, then arrange the four articles.

Answer 1

The Correct Option is B

Step 1: Count the total ways to pick 4 positions out of 8.
The 8 persons stand in a fixed row of 8 positions, and we must choose 4 of them to receive an article.
Total ways to choose 4 positions \( = \binom{8}{4} = 70 \).

Step 2: Identify the forbidden cases.
No four neighbouring persons receive an article rules out the case where the 4 chosen positions are all consecutive.
In a row of 8 positions, the consecutive blocks of 4 start at position 1, 2, 3, 4 or 5, giving the sets \( \{1,2,3,4\}, \{2,3,4,5\}, \{3,4,5,6\}, \{4,5,6,7\}, \{5,6,7,8\} \).
That is 5 forbidden position sets.

Step 3: Subtract the forbidden cases.
Valid ways to choose the 4 positions \( = 70 - 5 = 65 \).

Step 4: Arrange the 4 distinct articles among the chosen positions.
The 4 articles A, B, C and D are distinct, so they can be assigned to the 4 chosen positions in \( 4! = 24 \) ways.

Step 5: Multiply to get the final count.
Total ways \( = 65 \times 24 = 1560 \).

Final Answer:
There are 1560 valid ways to distribute the four articles. \[ \boxed{1560} \]