ShmoopTube

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Combinations, a la Shmoop. After a good, long workout at the gym, Clark

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decides that he's earned a hearty meal at ShmoopDonald's as a reward.

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<<ordering>>"One Big Mac, large fries, large Coke, and a baked apple pie, please. Oh, and

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make the Coke "diet." The second Clark gets his food, he feels a

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bit guilty. So he promises himself to only eat two out

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of four of the things he ordered. How many possible combinations of two items

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can Clark have? First of all, a combination is a selection

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of objects in which the order doesn't matter. In Clark's case, it doesn't matter if he has

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the Big shmoop and soda or soda and Big Shmoop... no matter the order, those calories are still

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getting in there. To determine the number of two-item combinations...

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...we can use the following formula: n choose r equals n factorial divided by n minus r

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factorial times r factorial. ...where n is the number of total items we

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have, which in this case is 4... ...and r is the number of items we want in

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a combination. So now let's plug in what we know.

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We get 4 choose 2 equals 4 factorial over 4 minus 2 factorial times 2 factorial.

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Remember... that exclamation mark represents factorial... or the product of an integer

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and all the integers below it. So, 4 factorial equals 4 times 3 times 2 times

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1... which equals 24. Now let's work with the bottom.

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4 minus 2 is 2... 2 factorial is 2 times 1, which is just 2.

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So 2 factorial times 2 factorial is 2 times 2... which is just 4.

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24 divided by 4 is 6. So the number of 2-item combinations Clark can have is...6!

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Clark may still be erasing much of his hard work at the gym...

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...but at least he'll be giving his jaw muscles

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a workout.

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