SOLUTION: Belmont Records sends a disk jockey 10 new CD releases for possible use. In how many ways can the disk jockey select no more than 8 CDs?

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Question 1149050: Belmont Records sends a disk jockey 10 new CD releases for possible use. In how many ways can the disk jockey select no more than 8 CDs?

Found 2 solutions by VFBundy, ikleyn:
Answer by VFBundy(438) About Me  (Show Source):
You can put this solution on YOUR website!
Selecting "no more than 8 CDs" is not only the number of ways he can choose 8 CDs, but also the number of ways he can choose 7,6,5,4,3,2, and 1 CD.

Ways to choose 8 CDs: 10C8 = 10%21%2F%288%21%2A2%21%29 = 45
Ways to choose 7 CDs: 10C7 = 10%21%2F%287%21%2A3%21%29 = 120
Ways to choose 6 CDs: 10C6 = 10%21%2F%286%21%2A4%21%29 = 210
Ways to choose 5 CDs: 10C5 = 10%21%2F%285%21%2A5%21%29 = 252
Ways to choose 4 CDs: 10C4 = 10%21%2F%284%21%2A6%21%29 = 210
Ways to choose 3 CDs: 10C3 = 10%21%2F%283%21%2A7%21%29 = 120
Ways to choose 2 CDs: 10C2 = 10%21%2F%282%21%2A8%21%29 = 45
Ways to choose 1 CD: 10C1 = 10%21%2F%281%21%2A9%21%29 = 10

45 + 120 + 210 + 252 + 210 + 120 + 45 + 10 = 1012

Answer by ikleyn(52756) About Me  (Show Source):
You can put this solution on YOUR website!
.

My notice to the solution of other tutor: 
    Formally, "to select no more than 8 of 10" includes "to select 0 (zero) //i.e. NOTHING", too.

So, including the choice "to select 0 (zero) // i.e. NOTHING", the answer is 2013.



            The other solution is possible,  too. 

(1)  "The number of ways to select no more than 8 of 10" is the COMPLEMENT to 2%5E10 of "the number of ways to select 9 or 10 from 10".



(2)  The number of ways to select 9  or 10  from 10  is  C%5B10%5D%5E9 + C%5B10%5D%5E10 = 10 + 1 = 11.



(3)  Therefore, the ANSWER to the problem's question is  


         The number of ways to select no more than 8 of 10 = 2%5E10 - 11 = 1024 - 11 = 1013.


When you know this  COMPLEMENTARY  property,  it may help you to reduce your calculation  SIGNIFICANTLY  in many cases.