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Problem 590

AMC 10/12, early questions

Combinatorics Difficulty 3.7 Multiple choice CEMC Pascal · Canada · 2022

Miyuki texted a six-digit integer to Greer. Two of the digits of
the six-digit integer were 3s. Unfortunately, the two 3s that Miyuki
texted did not appear and Greer instead received the four-digit integer
2022. The number of possible six-digit integers that Miyuki could have
texted is

Pick one

Official solution

The six-digit integer that Miyuki sent included the digits 2022
in that order along with two 3s.

If the two 3s were consecutive digits, there are 5 possible integers:
$332022 \qquad 233022 \qquad 203322 \qquad 202332 \qquad 202233$ We can think about the pair of 3s being
moved from left to right through the integer.

If the two 3s are not consecutive digits, there are 10 possible pairs of
locations for the 3s: 1st/3rd, 1st/4th, 1st/5th, 1st/6th, 2nd/4th,
2nd/5th, 2nd/6th, 3rd/5th, 3rd/6th, 4th/6th.

These give the following integers: $323022 \quad 320322 \quad 320232 \quad 320223 \quad 230322 \quad 230232 \quad 230223 \quad 203232 \quad 203223 \quad 202323$ (We can think
about moving the leftmost 3 from left to right through the integer and
finding all of the possible locations for the second 3.)

In total, there are thus $5 + 10 =
15$ possible six-digit integers that Miyuki could have
texted.

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.