Question

Jose is applying to college. He receives information on 10 different colleges. He will apply to all of those he likes. He may like any combination of them, or none of them, but he will not apply to all of them. How many possibilities are there for the set of colleges that he applies to?

Answer 1

Jose is applying to college. He receives information on 10 different colleges.

Case 1 : He applies to none of the colleges. This can be done in 1 way

Case 2 : He applies to 1 of the colleges. This can be done in 10 ways

Case 3 : He applies to 2 of the colleges. This can be done in ways becuase he needs to select 2 colleges from 10. = 10! / 2! (10 - 2)! = 45

Case 4 : He applies to 3 of the colleges. This can be done in ways becuase he needs to select 3 colleges from 10.    = 10! / 3! (10 - 3)! = 120

Case 5 : He applies to 4 of the colleges. This can be done in ways becuase he needs to select 4 colleges from 10.    = 10! / 4! (10 - 4)! = 210

Case 6 : He applies to 5 of the colleges. This can be done in ways becuase he needs to select 5 colleges from 10.    = 10! / 5! (10 - 5)! = 252

Case 6 : He applies to 6 of the colleges. This can be done in ways becuase he needs to select 6 colleges from 10.    = 10! / 6! (10 - 6)! = 210

Case 7 : He applies to 7 of the colleges. This can be done in ways becuase he needs to select 7 colleges from 10.    = 10! / 7! (10 - 7)! = 120

Case 8 : He applies to 8 of the colleges. This can be done in ways becuase he needs to select 8 colleges from 10.    = 10! / 8! (10 - 8)! = 45

Case 9 : He applies to 9 of the colleges. This can be done in ways becuase he needs to select 9 colleges from 10.    = 10! / 9! (10 - 9)! = 10

So the total number of ways he can apply = 1 + 10 + 45 + 120 + 210 + 252 + 210 + 120 + 45 + 10

= 1023 ways