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Combination probability

OCR S2 Jan 12
5(i) Six prizes are allocated, using random numbers, to a group of 12 girls and 8 boys. Calculate the probability that exactly 4 of the prizes are allocated to girls if
(a) the same child may win more than one prize, [2]
(b) no child may win more than one prize. [2]

Not sure about (b)
12C4 x 8C2 / 20C6 is the answer, but not sure how the 20C6 works to ensure that no child can appear any more than once in the 12C4 x 8C2 selection.

Reply 1

atsruser

Original post by xlaser31

Not sure about (b)
12C4 x 8C2 / 20C6 is the answer, but not sure how the 20C6 works to ensure that no child can appear any more than once in the 12C4 x 8C2 selection.

You want to find

\frac{\text{\# ways of allocating 6 prizes to 4 Gs,2 Bs}}{\text{\# ways of allocating 6 prizes among 20 people in any way}}

.

The 20C6 counts the bottom of this fraction. The 12C4 counts the number of ways that 4 prizes can be allocated among 12 girls without giving more than one prize to any one girl, and similarly for the 8C2 amongst the boys.