probibilty with classes

Question

Maya, a college freshman, is deciding on classes for her first term. She can choose to take some or all of the seven classes from the list below. She must take at least one of them. Astronomy, Biology, Chemistry, Dance, Economics, Geography, History In how many different ways can Maya choose the classes to take? (

Answer 1

Hi Lilly, here is a suggestion for how to approach this problem. I will use the notation C(n, k) to mean the number of ways to choose k objects from n ("n choose k").

Maya can take either 1, 2, 3, 4, 5, 6, or 7 of the classes.

* If she takes 1 class, she has C(7, 1) choices for which class it is.

* If she takes 2 classes, she has C(7, 2) choices for which two classes they are.

* If she takes 3 classes, she has C(7, 3) choices for which three classes they are.

... and so on, up to:

* If she takes 7 classes, she has C(7, 7) choices for which seven classes they are.

So the total number of possible schedules for her is:

C(7, 1) + C(7, 2) + C(7, 3) + C(7, 4) + C(7, 5) + C(7, 6) + C(7, 7).

You can plug that into a calculator to compute, or you can use the binomial identity:

C(7, 0) + C(7, 1) + C(7, 2) + C(7, 3) + C(7, 4) + C(7, 5) + C(7, 6) + C(7, 7) = 2^7, which means

C(7, 1) + C(7, 2) + C(7, 3) + C(7, 4) + C(7, 5) + C(7, 6) + C(7, 7) = 2^7 - C(7, 0) = 128 - 1 = 127.

I hope this is helpful for you! Feel free to contact me for more help in probability!