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Math  /  Discrete

QuestionIn a card game using a standard 52-card deck, five cards are selected. In how many ways can three red cards and two black cards be selected?
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Studdy Solution

STEP 1

1. A standard deck has 52 cards, with 26 red cards and 26 black cards.
2. We are selecting a total of 5 cards: 3 red and 2 black.

STEP 2

1. Determine the number of ways to select 3 red cards.
2. Determine the number of ways to select 2 black cards.
3. Calculate the total number of ways to select the cards by multiplying the results from the previous steps.

STEP 3

Determine the number of ways to select 3 red cards from 26 red cards using the combination formula:
(nk)=n!k!(n−k)! \binom{n}{k} = \frac{n!}{k!(n-k)!}
(263)=26!3!(26−3)! \binom{26}{3} = \frac{26!}{3!(26-3)!}
Calculate:
(263)=26×25×243×2×1=2600 \binom{26}{3} = \frac{26 \times 25 \times 24}{3 \times 2 \times 1} = 2600

STEP 4

Determine the number of ways to select 2 black cards from 26 black cards using the combination formula:
(262)=26!2!(26−2)! \binom{26}{2} = \frac{26!}{2!(26-2)!}
Calculate:
(262)=26×252×1=325 \binom{26}{2} = \frac{26 \times 25}{2 \times 1} = 325

STEP 5

Calculate the total number of ways to select the cards by multiplying the results from Step 1 and Step 2:
Total Ways=(263)×(262) \text{Total Ways} = \binom{26}{3} \times \binom{26}{2}
Total Ways=2600×325=845000 \text{Total Ways} = 2600 \times 325 = 845000
The total number of ways to select three red cards and two black cards is:
84500 \boxed{84500}

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