Inside Collection (Course): ENGR 2113 ECE Math
Summary: An introduction to combinations.
Let’s start once again with a deck of 52 cards. But this time, let’s deal out a poker hand (5 cards). How many possible poker hands are there?
At first glance, this seems like a minor variation on the Solitaire question above. The only real difference is that there are five cards instead of six. But in face, there is a more important difference: order does not matter. We do not want to count “Ace-King-Queen-Jack-Ten of spades” and “Ten-Jack-Queen-King-Ace of spades” separately; they are the same poker hand.
To approach such question, we begin with the permutations question: how many possible poker hands are there, if order does matter?
The key insight is that this second question—“How many different times are we counting, for instance, Ace-King-Queen-Jack-Ten of spades?”—is itself a permutations question! It is the same as the question “How many different ways can these five cards be rearranged in a hand?” There are five possibilities for the first card; for each of these, four for the second; and so on. The answer is 5! which is 120. So, since we have counted every possible hand 120 times, we divide our earlier result by 120 to find that there are
This question—“how many different 5-card hands can be made from 52 cards?”—turns out to have a surprisingly large number of applications. Consider the following questions:
Each of these is a combinations question, and can be answered exactly like our card scenario. Because this type of question comes up in so many different contexts, it is given a special name and symbol. The last question would be referred to as “300 choose 10” and written
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