Elliot has x two-coloured counters, 5 showing red. He turns over two different counters chosen at random. Given the probability that 5 still show red is 19/32, find x with algebra.

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Accepted Answer

There are 5 red and x − 5 green counters The count stays at 5 only if one red is turned over and one green is turned over P(red then green) = (5/x) × ((x − 4)/x), because turning the red over makes x − 4 green P(green then red) = ((x − 5)/x) × (6/x), because turning the green over makes 6 red Adding: (5(x − 4) + 6(x − 5))/x² = (11x − 50)/x² So (11x − 50)/x² = 19/32, which gives 32(11x − 50) = 19x² 352x − 1600 = 19x², so 19x² − 352x + 1600 = 0 Factorise: (x − 8)(19x − 200) = 0, so x = 8 or x = 200/19 x must be a whole number, so x = 8 Answer: 8

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