Question

1. a 7 - card hand is dealt from a 52 - card deck. which expression represents the number of possible hands? a. 52 , 7 b. 52! c. 52p7 d. 52c7

Explanation:

Step1: Understand permutation and combination

Permutation \(nPr=\frac{n!}{(n - r)!}\) (order matters). Combination \(nCr=\frac{n!}{r!(n - r)!}\) (order does not matter).

Step2: Analyze the card - hand problem

In a 7 - card hand from a 52 - card deck, the order of the cards in the hand does not matter.

Step3: Evaluate each option

• Option A: \(52\times7\) is a simple multiplication and not a combinatorial formula.
• Option B: \(52!\) is the factorial of 52, which is not relevant for choosing 7 cards.
• Option C: \(_{52}P_7=\frac{52!}{(52 - 7)!}\) (order matters in permutation, which is not the case for a card - hand).
• Option D: \(_{52}C_7=\frac{52!}{7!(52 - 7)!}\) (order does not matter in combination, which is correct for counting the number of 7 - card hands).

Answer:

D. \(52C7\)