SOLUTION: 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

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): You can put this solution on YOUR website!
Can't have 1 black button, that means you'd have 8 brown buttons but you only have 7.
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Set up a table.
Let X be the number of black buttons chosen.
Let Y be the number of brown buttons chosen.
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So sum the last column in the rows from X=2 to X=5,

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