How do we play poker without those cards?
Briony takes a standard pack of 52 cards and throws some cards away. However, she makes sure she keeps all four aces among the remaining cards. She then selects four cards at random from these remaining cards. If the probability of her selecting the four aces is , how many cards did she throw away?
The answer is 38.
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1 solution
Assume that Briony has k cards left in the pack after she threw some cards away. Then, there are 2 4 k ( k − 1 ) ( k − 2 ) ( k − 3 ) different ways of selecting four cards. Since there is only one way of picking up four aces, the probability of such an event is k ( k − 1 ) ( k − 2 ) ( k − 3 ) 2 4 . Hence
k ( k − 1 ) ( k − 2 ) ( k − 3 ) 2 4 = 1 0 0 1 1
Therefore k ( k − 1 ) ( k − 2 ) ( k − 3 ) = 2 4 × 1 0 0 1 . There is only one value of k that satisfies this equation, and the fact that 1 0 0 1 = 7 × 1 1 × 1 3 helps to quickly find it: k = 1 4 . Thus the answer is 5 2 − 1 4 = 3 8 .