Question

In the game of poker played with an ordinary deck of 52 cards various five-card holdings are given special names. The name "three of a kind" is reserved for a holding that meets the following rule: Three cards of the same denomination and two other cards of different denominations. The number of distinct "three of a kind" that can be drawn from a 52-card deck can be calculated with the following product expression: 2 (13) (1) (2) (1) 1. Calculate the value of the expression Enter the value as a number with the usual comma separators. For example, you would enter the number ten thousand as 10,000.

Answer 1

The expression provided (2 (13) (1) (2) (1) 1) seems incorrect for calculating "three of a kind" poker hands. The correct formula is 13 × (4 choose 3) × (12 choose 2) × 4 × 4, which equals 54,912 possible hands.

Step-by-step explanation:

The calculation of the number of distinct "three of a kind" poker hands from a 52-card deck involves a mathematical expression. To find this, we consider that we need to pick three cards of the same denomination and two other cards of different denominations. The expression provided is 2 (13) (1) (2) (1) 1, which simplifies to 2 × 13 × 1 × 2 × 1 × 1 or 52. However, this seems to be an incorrect expression for calculating "three of a kind". The correct expression should calculate choosing one denomination out of thirteen for the three cards, combined with choosing three out of four suits for these cards. Additionally, we need to choose two different denominations from the remaining twelve and then pick one out of four suits for each of these two cards. So the correct expression would be 13 × (4 choose 3) × (12 choose 2) × 4 × 4 which gives 13 × 4 × 66 × 4 × 4 = 54,912. This is the number of "three of a kind" poker hands possible in a well-shuffled deck.