Question

The number of ways, in which the letters A, B, C, D, E can be placed in the 8 boxes of the figure below so that no row remains empty and at most one letter can be placed in a box, is:

• A. 5880
• B. 960
• C. 840
• D. 5760

Hint:

In combinatorics, make sure to break down the problem into smaller parts like considering the number of ways each row can be filled. Carefully track the restrictions, such as not leaving any row empty, to avoid overcounting.

Answer 1

The Correct Option is D

Arrange the 8 boxes into three rows, designated as \( R_1, R_2, \) and \( R_3 \) for the 1st, 2nd, and 3rd rows, respectively. The total number of arrangements is calculated as follows: \( \text{Total} = \left[ (\text{All in } R_1 \text{ and } R_3) + (\text{All in } R_2 \text{ and } R_3) + (\text{All in } R_1 \text{ and } R_2) \right] \) \( = 8C5 \times 5! - \left[ \text{(ways to place in 1st and 2nd row)} + \text{(ways to place in 3rd row)} \right] \) \( = \left| (56-1) \times 6 \right| = 120 \times 48 = 5760 \). Therefore, there are \( 5760 \) distinct ways to arrange the letters.