Question

1. how many combinations of 3 playing cards (from a 52-card deck) exist?

1,649
22,100
52³
25,000

Explanation:

Step1: Recall Combination Formula

The formula for combinations is \( C(n, k) = \frac{n!}{k!(n - k)!} \), where \( n = 52 \) (total cards) and \( k = 3 \) (cards to choose).

Step2: Calculate Factorials

First, compute \( n! = 52! \), \( k! = 3! \), and \( (n - k)! = 49! \). Then, \( C(52, 3) = \frac{52!}{3! \times 49!} \). Simplify \( \frac{52!}{49!} = 52 \times 51 \times 50 \), so \( C(52, 3) = \frac{52 \times 51 \times 50}{3 \times 2 \times 1} \).

Step3: Perform Arithmetic

Calculate numerator: \( 52 \times 51 = 2652 \), \( 2652 \times 50 = 132600 \). Denominator: \( 3 \times 2 \times 1 = 6 \). Then, \( \frac{132600}{6} = 22100 \).

Answer:

22,100 (the option with "22,100")