if 5 people are randomly selected, find the probability that no 2 of them have the same birthday. ignore leap years. the probability that 5 people have different birthdays is (round to three decimal places as needed.)

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Accepted Answer

# Explanation: ## Step1: The first person can have a birthday in 365 ways. The second person also has 365 ways (since we are considering all possible birthdays), the third person 365 ways, the fourth person 365 ways and the fifth person 365 ways. So the total number of ways 5 people can have birthdays is $n = 365\times365\times365\times365\times365=365^{5}$ ## Step2: The first person can have a birthday in 365 ways. The second person must have a different birthday from the first, so 364 ways. The third person must have a different birthday from the first two, so 363 ways. The fourth person 362 ways and the fifth person 361 ways. So the number of ways 5 people have different birthdays is $m=365\times364\times363\times362\times361$ ## Step3: The probability $P$ that no two of the 5 people have the same birthday is $P=\frac{m}{n}=\frac{365\times364\times363\times362\times361}{365^{5}}$ $$ \begin{align*} P&=\frac{365!}{(365 - 5)!\times365^{5}}\ &=\frac{365\times364\times363\times362\times361}{365\times365\times365\times365\times365}\ &=\frac{364\times363\times362\times361}{365^{4}}\ \end{align*} $$ $$ \begin{align*} \frac{364\times363\times362\times361}{365^{4}}&=\frac{364\times363\times362\times361}{(365)^{4}}\ &=\frac{364\times363\times362\times361}{177975750625}\ &=\frac{364\times363\times 130082}{177975750625}\ &=\frac{364\times47219766}{177975750625}\ &=\frac{17287994724}{177975750625}\ &\approx0.973 \end{align*} $$ # Answer: $0.973$

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Question

Explanation:

Step1: <Calculate total possible birthday combinations>

Step2: <Calculate number of ways 5 people have different birthdays>

Step3: <Calculate the probability>

Answer: