“ You are dealt five cards from a standard deck of 52 cards. How many of these hands have at least one queen and at least one spade. ” He breaks this into two cases. In the first case, he looks at hands which contain the queen of spades Q♠. There is 1 way of choosing the Queen of Spades, and 51C4 ways of choosing the other four cards. In the second case, he will start by choosing a non-spade queen (3 ways) and a non-queen spade (12 ways). There are 50 cards remaining, so he also throws away the Queen of Spades so that he doesn’t overlap case 1. This gives 49C3 ways of choosing the last three cards. He concludes that the answer is 51C4 + (3 · 12 · 49C3). His answer is incorrect. Explain why, and give the correct answer.

Question

“ You are dealt five cards from a standard deck of 52 cards. How many of these hands have at least one queen and at least one spade. ”

He breaks this into two cases. In the first case, he looks at hands which contain the queen of spades Q♠. There is 1 way of choosing the Queen of Spades, and 51C4 ways of choosing the other four cards. In the second case, he will start by choosing a non-spade queen (3 ways) and a non-queen spade (12 ways). There are 50 cards remaining, so he also throws away the Queen of Spades so that he doesn’t overlap case 1. This gives 49C3 ways of choosing the last three cards. He concludes that the answer is 51C4 + (3 · 12 · 49C3). His answer is incorrect. Explain why, and give the correct answer.

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