How many hands of 5 cards can be made from a standard deck of cards such that each hand contains exactly 1 red card and 4 black cards?
To find the number of hands of 5 cards that contain exactly 1 red card and 4 black cards, we can break down the problem into smaller steps:
Step 1: Determine the number of ways to choose 1 red card from the deck.
A standard deck of cards contains 26 black cards and 26 red cards. Since we want to choose only 1 red card, there are 26 possibilities.
Step 2: Determine the number of ways to choose 4 black cards from the remaining deck.
After selecting 1 red card, there are 51 cards remaining in the deck, of which 26 are black. We need to choose 4 black cards from this set, which can be done in 26 choose 4 ways.
Step 3: Calculate the total number of hands.
The total number of hands can be obtained by multiplying the results from Step 1 and Step 2 together, since these steps are independent. Therefore, the total number of hands is:
26 * (26 choose 4)
To simplify this calculation, we can use the formula for combinations, which states that n choose r is equal to n! / (r! * (n - r)!), where n is the total number of items and r is the number of items to be selected.
Plugging in our values:
Total number of hands = 26 * (26 choose 4)
= 26 * (26! / (4! * (26 - 4)!)
Simplifying further using factorial notation:
Total number of hands = 26 * (26! / (4! * 22!))
Calculating the factorial terms:
Total number of hands = 26 * (26 * 25 * 24 * 23) / (4 * 3 * 2 * 1)
Finally, evaluating the expression:
Total number of hands = 26 * 149,600 / 24
= 26 * 6,233.33
= 162,200
Therefore, there are a total of 162,200 hands of 5 cards that can be made from a standard deck of cards such that each hand contains exactly 1 red card and 4 black cards.