Question 1070134: Problem: A jar contains seven black buttons and seven brown buttons. If nine buttons are picked at random, what is the probability that at most five of them are black?
My Attempt: I found the total ways to pick 9 out of 14 using the combination formula, 2002. I then using combination listed the possible ways to pick 1,2,3,4 and 5 black buttons.
1:49
2:441
3:1125
4:1125
5:441
Adding these gives me a number greater than 2002...but probability must be less than 1... Where might have I went wrong? I know the correct answer is 124/143 due to the answer key. Thanks in Advance!
Found 2 solutions by Boreal, Fombitz:
Answer by Boreal(15235)   (Show Source):
You can put this solution on YOUR website! The way to do this is to first recognize that 2 is the least number of black buttons, because one will pick 9, and the other 7 must be brown. If one picks 1, then it is not possible to have 8 brown buttons, only 7.
So the first term in the numerator is 7C2*7C7, or 21.
7C3*7C6 is the second possibility and 35*7=245
7C4*7C5 is the third possibility and 35*21=735
7C5*7C4 is the last possibility and is also 735
Those 4 add to 1736
1736/2002=124/143
Notice how the combinatorials add to 14C9; the first (top) part adds to 14, the last (bottom) part to 9.
Answer by Fombitz(32388)  (Show Source):
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