1. **Stating the problem:** We have a deck of 52 cards, and we want to find two things: - How many cards can be chosen from the deck? - How many cards must be chosen to guarantee at least three gilts (assuming "gilts" means a specific type of card, e.g., a suit or rank). 2. **Understanding the deck:** A standard deck has 52 cards divided into 4 suits, each with 13 cards. If "gilts" refers to one suit, there are 13 gilts. 3. **How many cards can be chosen?** You can choose any number of cards from 0 up to 52. So the number of cards that can be chosen ranges from 0 to 52. 4. **How many cards must be chosen to have at least three gilts?** We use the pigeonhole principle here. - There are 13 gilts. - To guarantee at least 3 gilts, consider the worst case: you pick as many cards as possible without getting 3 gilts. 5. **Worst case scenario:** - You can pick at most 2 gilts without reaching 3. - You can pick all non-gilt cards: 52 - 13 = 39 cards. 6. **Total cards picked without 3 gilts:** $$ 2 \text{ (gilts)} + 39 \text{ (non-gilts)} = 41 $$ 7. **Therefore, to guarantee at least 3 gilts, you must pick:** $$ 41 + 1 = 42 $$ **Final answers:** - Number of cards that can be chosen: any number from 0 to 52. - Number of cards to guarantee at least 3 gilts: 42.