Choosing cards (combinations)

Goal: Count selections where order does not matter, using C(n, k) = P(n, k) ÷ k!.

Remember: Divide ordered counts by k! to remove repeated orderings of the same set.

1. 1.How many ways are there to choose 2 cards from 5 different cards?

2. 2.How many ways are there to choose 2 hearts from the 13 hearts?

3. 3.Show that there are 2,598,960 five-card hands. Write the ordered count, then divide.

4. 4.How many five-card hands contain no hearts at all?

5. 5.How many five-card hands have all five cards in the same suit (including runs)?

6. 6.How many five-card hands contain all four aces?

7. 7.Without calculating, explain why C(52, 5) = C(52, 47).

8. 8.Count the five-card hands with exactly one pair: 13 ranks × C(4, 2) suit pairs × C(12, 3) other ranks × 4³ suit choices.