a deck of playing cards has four suits, with thirteen cards in each suit consisting of the numbers 2 through…

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.

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.

Answer

Explanation:

Step1: Determine total - possible outcomes

When choosing a hand of 5 cards from a deck of 52 cards, the order of the cards in the hand does not matter. The formula for combinations is ${n}C{r}=\frac{n!}{r!(n - r)!}$, where $n = 52$ and $r = 5$. So the total number of possible outcomes is ${52}C{5}=\frac{52!}{5!(52 - 5)!}=\frac{52!}{5!47!}$. The formula for permutations ${n}P{r}=\frac{n!}{(n - r)!}$ is used when order matters, so the total possible outcomes are found using ${52}C{5}$, not ${52}P{5}$.

Step2: Calculate probability of two - diamonds and three - hearts

The number of ways to choose 2 diamonds out of 13 is ${13}C{2}=\frac{13!}{2!(13 - 2)!}=\frac{13\times12}{2\times1}=78$. The number of ways to choose 3 hearts out of 13 is ${13}C{3}=\frac{13!}{3!(13 - 3)!}=\frac{13\times12\times11}{3\times2\times1}=286$. The number of ways to choose 5 - card hands with 2 diamonds and 3 hearts is ${13}C{2}\times_{13}C_{3}=78\times286 = 22308$. The total number of 5 - card hands is ${52}C{5}=2598960$. The probability $P=\frac{{13}C{2}\times_{13}C_{3}}{{52}C{5}}=\frac{22308}{2598960}\approx0.0086$.

Step3: Calculate probability of five - spades

The number of ways to choose 5 spades out of 13 is ${13}C{5}=\frac{13!}{5!(13 - 5)!}=\frac{13\times12\times11\times10\times9}{5\times4\times3\times2\times1}=1287$. The probability $P=\frac{{13}C{5}}{{52}C{5}}=\frac{1287}{2598960}\approx0.0005$.

Step4: Calculate probability of five - clubs

The number of ways to choose 5 clubs out of 13 is ${13}C{5}=1287$. The probability $P=\frac{{13}C{5}}{{52}C{5}}=\frac{1287}{2598960}\approx0.0005$.

Answer:

The total possible outcomes can be found using ${52}C{5}$, The probability of choosing five clubs is roughly 0.0005.