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?

   Answer: 10

   Working: C(5, 2) = (5 × 4) ÷ (2 × 1)

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

   Answer: 78

   Working: C(13, 2) = (13 × 12) ÷ 2

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

   Answer: 52 × 51 × 50 × 49 × 48 = 311,875,200 ordered deals; each hand appears in 5! = 120 orders; 311,875,200 ÷ 120 = 2,598,960.

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

   Answer: 575,757

   Working: C(39, 5): all five cards come from the 39 non-hearts

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

   Answer: 5,148

   Working: 4 suits × C(13, 5) = 4 × 1,287

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

   Answer: 48

   Working: The four aces plus one of the other 48 cards: C(4, 4) × C(48, 1)

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

   Answer: Choosing the 5 cards to keep is the same decision as choosing the 47 to leave out.

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.

   Answer: 1,098,240

   Working: 13 × 6 × 220 × 64