AMC 10 Weekly Practice Round 3

Complete problem set with solutions and individual problem pages

Problem 7 Medium

Five people A, B, C and two others stand in a row, with A not at either end, and exactly two people standing between B and C. How many different arrangements are there?

  • A.

    20

  • B.

    16

  • C.

    12

  • D.

    8

  • E.

    6

Answer:B

Since there are exactly 2 people between B and C, the block consisting of B, C and the two people between them must occupy either the first four positions or the last four positions.

 

1. When the block occupies the first four positions, the last position remains free. In this case, A must be placed between B and C.

 

Arranging B and C: _{2}P_{2} ways.

Placing A between them: _{2}P_{1} way.

Arranging the other two people in the remaining two positions: _{2}P_{2} ways.

 

Thus, the number of arrangements is

_{2}P_{2}\times _{2}P_{1}\times _{2}P_{2} = 8.

 

2. When the block occupies the last four positions, the first position remains free. Similarly, A must be placed between B and C.

 

Arranging B and C: _{2}P_{2} ways.

Placing A between them: _{2}P_{1} way.

Arranging the other two people in the remaining two positions: _{2}P_{2} ways.

 

Thus, the number of arrangements is

_{2}P_{2}\times _{2}P_{1}\times _{2}P_{2} = 8.

 

By the Addition Principle of Counting, the total number of arrangements is

8 + 8 = 16.

 

Therefore, the answer is \text{B}.

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