Pairs game

Task number: 4455

How many different starting positions does the famous Pairs game have?

In other words, how many ways can 64 cards containing 32 identical pairs be arranged in a square \(8 \times 8\)?

  • Solution

    If we differentiate all cards, e.g. by numbers from 1 to 64, we would get \(64!\) possible (physical/numbered) placements.

    In a single starting position, each of the 32 pairs of identical cards can be swapped without changing the starting position.

    In other words, one unique starting position corresponds to \(2^{32}\) different (physical/numbered) placements.

    We get the number we are looking for by dividing the number of all placements by the number of placements that correspond to a single starting position.

  • Answer

    There are \(\frac{64!}{2^{32}}\doteq 2.95\,\cdot\,10^{79}\) different card layouts.

Difficulty level: Easy task (using definitions and simple reasoning)
Reasoning task
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