Question 143040
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The 7 even numbered balls are 2,4,6,8,10,12,14
The 8 odd numbered balls are  1,3,5,7,9,11,13,15.

There will be an odd sum is there is an odd number of
odd numbered balls, and an even sum if there is an even
number of odd balls.  So the cases of an odd number of
odd numbered balls:

N(1 odd numbered ball & 5 even numbered balls) = C(8,1)*C(7,5) = 8*21  =  168
N(3 odd numbered ball & 3 even numbered balls) = C(8,3)*C(7,3) = 56*35 = 1960
N(5 odd numbered ball & 1 even numbered balls) = C(8,5)*C(7,1) = 56*7  =  392
                                                                 Total = 2520

The total number of choices of any 6 balls out of the 15 is C(15,6) = 5005  

So the desired probability = 2520/5005 = 72/143

Edwin</pre>