Question

in how many ways can we select four cards of an ordinary pack of playing cards so that exactly three of them are of the same denomination.

Answer 1

2496

Answer 2

To select four cards from an ordinary pack of playing cards such that exactly three of them are of the same denomination, we follow these steps:

1. Choose the denomination for the three cards: There are 13 possible denominations (Ace, 2, ..., King). Number of ways to choose this denomination = $\binom{13}{1} = 13$.

2. Choose 3 cards from the 4 cards of the selected denomination: For the chosen denomination, there are 4 cards (one of each suit). We need to select 3 of these 4 cards. Number of ways to choose these 3 cards = $\binom{4}{3} = 4$.

3. Choose the denomination for the fourth card: The fourth card must be of a different denomination than the one selected in step 1 to ensure "exactly three" are of the same denomination. Since one denomination has already been chosen, there are 12 remaining denominations. Number of ways to choose this new denomination = $\binom{12}{1} = 12$.

4. Choose 1 card from the 4 cards of the newly selected denomination: For this new denomination, there are 4 cards (one of each suit). We need to select 1 of these 4 cards. Number of ways to choose this 1 card = $\binom{4}{1} = 4$.

To find the total number of ways, we multiply the number of ways for each step (by the multiplication principle):

Total number of ways = (Ways to choose denomination for 3 cards) $\times$ (Ways to choose 3 cards from that denomination) $\times$ (Ways to choose denomination for 1 card) $\times$ (Ways to choose 1 card from that denomination)

Total number of ways = $\binom{13}{1} \times \binom{4}{3} \times \binom{12}{1} \times \binom{4}{1}$ Total number of ways = $13 \times 4 \times 12 \times 4$ Total number of ways = $52 \times 48$ Total number of ways = $2496$