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Challenge 272: Eggcup Match-up

I have a bag containing 20 wooden eggs. 10 of them are painted green and 10 are painted yellow, but otherwise they are identical.

I also have 3 green eggcups and 3 yellow eggcups.

If I pick eggs at random from the bag to put in each eggcup, what is the probability that each eggcup and its egg are of matching colours?

If I have 10 green eggcups and 10 yellow eggcups and place eggs in them in the same way, what is the probability that they are all placed so their colours match?

Can you generalise to give an expression for the probability that all the eggs and eggcups match when there are 2n eggs (n of each colour) and 2r eggcups (r of each colour), where r does not exceed n ?

For writing down your answers, you may find it useful to know that the notation n! stands for

n(n-1)(n-2)…3.2.1 – so, for example 4! = 4 x 3 x 2 x 1= 24

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