“ 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.
Permutations and Combinations
If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.
Counting Theory
The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.
“ 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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