1983 AIME Problem 10

Attempt Problem 10 of the 1983 AIME below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1983 AIME solutions, or check the answer key.

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10.

The numbers 1447,1447, 1005,1005, and 12311231 have something in common: each is a four-digit number beginning with 11 that has exactly two identical digits. How many such numbers are there?

Answer: 432
Concepts:digitscaseworkmultiplication principle
Difficulty rating: 1900
Small Hint:

Separate the case where the repeated digit is 11 from the case where it is not

Big Hint:

In each case, choose the repeated digit’s positions before choosing the remaining distinct digit

Solution:

If 11 is the repeated digit, exactly one of the last three positions contains 1.1. There are 33 choices for that position, 99 choices for the next digit, and 88 for the last digit, since those two digits must differ from each other and from 1.1. This gives 398=2163\cdot9\cdot8=216 numbers.

If a digit other than 11 is repeated, there are 99 choices for that digit, 33 ways to choose its two positions among the last three, and 88 choices for the remaining digit. This gives another 938=2169\cdot3\cdot8=216 numbers. The total is 216+216=432.216+216=432.

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