Questions: A deck of playing cards has four suits, with thirteen cards in each suit consisting of the numbers 2 through 10, a jack, a queen, a king, and an ace. The four suits are hearts, diamonds, spades, and clubs. A hand of five cards will be chosen at random. Which statements are true? Check all that apply. The total possible outcomes can be found using 52 C 5. The total possible outcomes can be found using 52 P 5. The probability of choosing two diamonds and three hearts is 0.089. The probability of choosing five spades is roughly 0.05 The probability of choosing five clubs is roughly 0.0005.

Solution Steps

The total number of ways to choose 5 cards from a deck of 52 cards is given by the combination formula: \[ {}_{52} \mathrm{C}_{5} = \frac{52!}{5!(52-5)!} \] This is because the order of the cards does not matter in a hand.

The first statement claims that the total possible outcomes can be found using \({}_{52} \mathrm{C}_{5}\). This is true, as explained in Step 1.

The second statement claims that the total possible outcomes can be found using \({}_{52} \mathrm{P}_{5}\). This is false because permutations (\({}_{52} \mathrm{P}_{5}\)) consider the order of selection, which is not relevant for counting the number of possible hands.

The third statement claims that the probability of choosing two diamonds and three hearts is 0.089. To verify this, calculate the probability as follows: \[ \text{Probability} = \frac{{}_{13} \mathrm{C}_{2} \times {}_{13} \mathrm{C}_{3}}{{}_{52} \mathrm{C}_{5}} \] Calculate \({}_{13} \mathrm{C}_{2}\) and \({}_{13} \mathrm{C}_{3}\): \[ {}_{13} \mathrm{C}_{2} = \frac{13!}{2!(13-2)!} = 78 \] \[ {}_{13} \mathrm{C}_{3} = \frac{13!}{3!(13-3)!} = 286 \] Now, calculate the probability: \[ \text{Probability} = \frac{78 \times 286}{2,598,960} \approx 0.0086 \] The given probability of 0.089 is incorrect.

Final Answer