SOLUTION: Jose is applying to college. he receives information on 9 different colleges. he will apply to all of those he likes. he may like none of them, all of them, or or any combination o

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He may apply to none of them. There is only one way he can do this.

He may apply to just one of them. There are 9 different colleges so there are 9 different ways he can do this.

He may apply to two of them. There are 9 colleges he can pick as his first choice, leaving 8 colleges from which he can pick his second choice. Therefore the number of ways he can do this is 9 X 8 = 72
However, since it doesn't matter which way round he picks the two colleges, the actual number of ways he can do this is 72/2 = 36

He may apply to three of them. The number of ways he can do this, assuming that the order in which he picks them isn't important is (9 X 8 X 7)/(3 X 2 X 1) which is equal to 84

He may apply to four of them. he can do this in (9 X 8 X 7 X 6)/(4 X 3 X 2 X 1) = 126

Applying to four is the same as NOT applying to five So the number of ways of picking four from nine is the same as the number of ways as picking five from nine. So the number of ways of applying to five colleges is, again, 126

Applying the same logic, there are 84 ways of applying to six colleges
There are 36 ways of applying to seven colleges

There are 9 ways of applying to eight colleges (leaving out one of the nine each time)

And of course, just one way of applying to all nine.
So the total number of ways is 1 + 9 + 36 + 84 + 126 + 126 + 84 + 36 + 9 + 1
which is 512.