2022 AMC 10 A

Complete problem set with solutions and individual problem pages

Problem 14 Easy

How many ways are there to split the integers 1 through 14 into 7 pairs such that in each pair, the greater number is at least 2 times the lesser number?

  • A.

    108

  • B.

    120

  • C.

    126

  • D.

    132

  • E.

    144

Answer:E

Integers from 8 to 14 have to be separated into differentt groups, because none of them could be the 'lesser number' in the pair to be the larger number. The lesser number candidates are 1,2,3,4,5,6,7.

7 \rightarrow 14

6 \rightarrow 12, 13 (let's go with 13)

5 \rightarrow 10, 11, 12 (let's go with 12)

4 \rightarrow 8, 9, 10, 11 (let's go with 11)

3 \rightarrow 8, 9, 10 (let's go with 10)

2 \rightarrow 8, 9 (let's go with 9)

1 \rightarrow 8

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