This math skill teaches how to count the distinct ways to order a set of cards, including identical ones (duplicates). Calculate the total permutations without considering duplicates, then divide by the permutations of the duplicates. Use factorial for both calculations.
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Counting ways to order a hand of cards with duplicates
Test your understanding of order cards with duplicates by practicing it! Work through the below exercises to use it in practice.
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How many distinct ways can we order these cards? We have 5 cards, but 3 of them are kings of hearts, so we'll need to account for those as duplicates
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We can calculate all the ways of ordering the cards, ignoring the duplicates, and then divide that by the number of ways that duplicates can be created
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Looking at all the possibilities, we have 5 options for the first card, and for each of those, 4 options for the second card, and so on.
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For the duplicates, we can calculate the ways in which 3 identical cards can be put in order, which is 3 for the first option, multiplied by 2 for the second, and so on.
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We can see that this is what a factorial is, so we have 5 factorial, divided by 3 factorial.
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Cancelling like terms on top and bottom, we are left with 5 times 4
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So we have a total of 20 distinct ways to order this set of cards