What is the number of ways of choosing 4 cards from a pack of 52 playing cards? In how many of these four cards are of the same suit?
Understand the Problem
The question is asking for the number of ways to choose 4 cards from a standard deck of 52 playing cards, specifically focusing on the scenario where all 4 chosen cards are of the same suit. It involves using combinatorial mathematics to calculate how many such combinations are possible based on the number of suits and cards in each suit.
Answer
The number of ways to choose 4 cards of the same suit is 2860.
Answer for screen readers
The total number of ways to choose 4 cards of the same suit from a standard deck of 52 playing cards is:
$$ 4 \times \frac{13!}{4!9!} = 4 \times 715 = 2860 $$
Steps to Solve
- Identify the total number of cards in each suit
In a standard deck, there are 4 suits: Diamonds, Spades, Hearts, and Clubs. Each suit has 13 cards.
- Calculate the ways to choose 4 cards from one suit
To find the number of ways to choose 4 cards from a suit, we use the combination formula:
$$ ^nC_r = \frac{n!}{r!(n-r)!} $$
In this case, $n = 13$ (the total number of cards in a suit) and $r = 4$ (the number of cards to choose).
- Apply the combination formula to one suit
Calculating the number of ways to choose 4 cards from 13 cards in one suit:
$$ ^{13}C_4 = \frac{13!}{4!(13-4)!} = \frac{13!}{4!9!} $$
- Calculate the number of ways for all suits
Since there are 4 suits, multiply the number of ways for one suit by the number of suits:
$$ \text{Total ways} = 4 \times ^{13}C_4 = 4 \times \frac{13!}{4!9!} $$
- Simplify the final expression
Now, simplify the expression to get the total number of ways to choose 4 cards of the same suit.
The total number of ways to choose 4 cards of the same suit from a standard deck of 52 playing cards is:
$$ 4 \times \frac{13!}{4!9!} = 4 \times 715 = 2860 $$
More Information
Choosing 4 cards from the same suit means that you are exclusively selecting cards from one category. Regardless of the suit you pick, the formula remains consistent due to the symmetry in the deck structure.
Tips
- Failing to multiply by the number of suits available (not considering all four suits).
- Misapplying the combination formula, especially in calculating $n!$ and $(n-r)!$.
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