Question

Some coins are to be put into 7 pockets so that each pocket contains at least one coin. At most 3 of the pockets are to contain the same number of coins, and no two of the remaining pockets are to contain an equal number of coins. What is the least possible number of coins needed for the pockets ?

• A. 17
• B. 7
• C. 28
• D. 13
• E. 22

Hint:

For minimization problems with distinct value constraints, always start allocating from the smallest possible allowed integer (e.g., 1) and increase incrementally strictly as rules require.

Answer 1

The Correct Option is A

Step 1: Determine the strategy for minimization.
To find the least possible number of coins, we must assign the smallest possible integers to the pockets.
Every pocket must have $\ge 1$ coin.

Step 2: Assign coins to the pockets.
We can have at most 3 pockets with the same number of coins. Let's assign the absolute minimum value (1 coin) to these 3 pockets: $1, 1, 1$.
The remaining 4 pockets must contain different numbers of coins, and none can equal 1 (otherwise we'd have more than 3 pockets with 1 coin).
To minimize the sum, assign the next smallest consecutive integers: 2, 3, 4, and 5.

Step 3: Calculate the total.
Sum = $1 + 1 + 1 + 2 + 3 + 4 + 5 = 17$.