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Order cards with duplicates

Counting ways to order a hand of cards with duplicates

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.

Learn order cards with duplicates

Teaching Transcript

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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

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

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.

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.

We can see that this is what a factorial is, so we have 5 factorial, divided by 3 factorial.

Cancelling like terms on top and bottom, we are left with 5 times 4

So we have a total of 20 distinct ways to order this set of cards

Skill Practice

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.