AMC 10 Weekly Practice Round 3

If the sum of the digits of a five-digit number is , then such numbers can be divided into three cases:

Case . The digits consist of four ’s and one .

Since the first digit cannot be , the must be in the first place and all other digits are , giving the number .

Thus, there is only possibility.

Case . The digits consist of three ’s, one , and one .

Since the first digit cannot be , the first digit must be either or . Choose one of them for the first place, and the other digit can occupy any of the remaining four positions. The rest are .

Thus, there are possibilities.

Case 3. The digits consist of two ’s and three ’s.

Since the first digit cannot be , the first digit must be . Among the remaining four places, choose two for ’s, and the rest are .

Thus, there are possibilities.

By the Addition Principle of Counting, the total number of such five-digit numbers is

Therefore, the answer is .