Question

a standard card deck has 52 cards. how many three - card hands are possible from a standard deck?
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Explanation:

Step1: Identify the problem type

This is a combination problem where we want to find the number of ways to choose 3 cards from a deck of 52 cards. The formula for combinations is \( C(n, k) = \frac{n!}{k!(n - k)!} \), where \( n = 52 \) (total number of cards) and \( k = 3 \) (number of cards in a hand).

Step2: Plug in the values into the combination formula

First, calculate the factorials. The formula becomes \( C(52, 3) = \frac{52!}{3!(52 - 3)!} = \frac{52!}{3! \times 49!} \).

Since \( n! = n \times (n - 1) \times \dots \times (n - k + 1) \times (n - k)! \), we can simplify \( 52! \) as \( 52 \times 51 \times 50 \times 49! \). Then the \( 49! \) terms cancel out, leaving \( C(52, 3) = \frac{52 \times 51 \times 50}{3 \times 2 \times 1} \).

Step3: Calculate the numerator and the denominator

Calculate the numerator: \( 52 \times 51 \times 50 = 52 \times 2550 = 132600 \).

Calculate the denominator: \( 3 \times 2 \times 1 = 6 \).

Step4: Divide the numerator by the denominator

\( \frac{132600}{6} = 22100 \).

Answer:

22100