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?

We want to minimise the total number of coins under these conditions:

7 pockets, each has at least 1 coin

At most 3 pockets can have the same number of coins

The remaining pockets must all have distinct numbers

To minimise total coins: we can use the smallest possible positive integers and allow exactly 3 pockets to have the same number (since “at most 3” → we use 3 to minimise total). The other 4 pockets must all be distinct and different from the repeated number.

Let the repeated number be 1 (smallest possible): so 3 pockets: 1, 1, 1

Now choose 4 distinct numbers, all different from 1: Next smallest distinct numbers: 2, 3, 4, 5

Hence, Total coins: 1 + 1 + 1 + 2 + 3 + 4 + 5 = 17