Easy Combination Possibilities Questions?
I like to know step by step solution not just solution. I would appreciate if you use one line for one step so that it does not look cluttered and difficult to read.
Please explain the "trick" with the combination "numbers", for example, if there are 3 choices, then what, and if there were 5 choices, then what. Thanks
1) There are 4 contestants in a bicycle race. In how many different orders can they cross the finish line? (I think in this one, we may repeat each subject in ranking)
2) Clayton has to read 3 books for English class. He can choose from a list of 5 books. How many different combinations of 3 books are possible? (In this one, we may not repeat the same book combinations?)
3) Carly is making cupcakes with nuts. She can choose between 4 flavors of cupcakes, 3 colors of frosting, and 2 kinds of nuts. She chooses one flavor, one color and one kind of nut. How many different cupcakes could be made?
1) How many different racers can come in first: 4.
Given a specific racer coming in first, 3 possible racers could come in second.
Given specific racers in first and second, there are 2 possibilities for third place.
There is only one possible racer for fourth place given the first through third positions.
Therefore, the total number of orderings is 4*3*2*1 = 24.
2) Clayton has 5 possibilities for the first book, 4 for the second, and 3 for the third book he chooses.
This gives 5*4*3 = 60 possible ordered ways to choose the books.
However, we don't care what order he chooses the books, thus we must divide 60 by the number of different ways to choose three particular books.
Given three particular books, there are three possibilities for the first choice, two for the second, and only one for the third choice.
Thus there are 3*2 = 6 different orderings for each choice of three books.
Therefore there are 60/6 = 10 choices of three books.
3) 4*3*2 = 24 cupcakes.
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