2025 AMC 10 A

Complete problem set with solutions and individual problem pages

Problem 16 Easy

There are three jars. Each of three coins is placed in one of the three jars, chosen at random and independently of the placements of the other coins. What is the expected number of coins in a jar with the most coins?

  • A.

    \frac { 4 } { 3 }

  • B.

    \frac { 1 3 } { 9 }

  • C.

    \frac { 5 } { 3 }

  • D.

    \frac { 1 7 } { 9 }

  • E.

    2

Answer:D

Total arrangements for placing 3 coins in 3 jars: 3^3 = 27.

If maximum coins per jar = 1: Each jar contains exactly one coin. Count: 3! = 6.

If maximum coins per jar = 3: One jar holds all coins. Count: 3.

Remaining arrangements (27 - 6 - 3 = 18): One jar has 2 coins, another has 1.

Expected maximum: \frac{6}{27} \times 1 + \frac{18}{27} \times 2 + \frac{3}{27} \times 3 = \frac{17}{9}

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