question archive 1) In a deck of cards there are two black suits {♠, ♣} and two red suits {♥, ♦}
1) In a deck of cards there are two black suits {♠, ♣} and two red suits {♥, ♦}
Subject:MathPrice: Bought3
1) In a deck of cards there are two black suits {♠, ♣} and two red suits {♥, ♦}. Suppose that Parker calculated the number of five card hands containing exactly three black cards, and exactly two red cards. His calculation appears in this grey box:
To count the number of five card hands with exactly three black cards, and exactly two red cards, we proceed as follow: 1. For the three black cards there are two suits {♠, ♣}, so we get 3C2 choices of suits.
2. For the two red cards, there are two suits {♥, ♦}, so we get 2C2 choices of suit.
3. We must then assign ranks to the cards. There are 13C5 choices for five ranks.
So, we get: (3C2)(2C2)(13C5) = 3,861
Unfortunately, Parker's calculation is incorrect! The true number is 845, 000.
(i) Identify the errors in his reasoning that led to this incorrect answer. For full marks, you must explain which steps are incorrect and why they are incorrect.
(ii) Parker's count is much smaller than the true number. There are 845, 000 − 3, 861 = 841, 139 valid arrangements which Parker's count did not include. Provide two examples to receive full marks.
(iii) find the formula which gives the correct number of five card hands containing exactly three black cards, and exactly two red cards. Explain your formula in at most one hundred and fifty words, like Parker did.
