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The Editor-in-Chief of the student newspaper was doing a final review of the articles submitted for the upcoming edition. Of the $7$ total articles submitted, $5$ were editorials. If she liked all the articles equally, and randomly selected $4$ articles to go on the front cover, what is the probability that all of them are editorials? Write your answer as a decimal rounded to four decimal places. ____

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Find probabilities using combinations and permutations

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Q. The Editor-in-Chief of the student newspaper was doing a final review of the articles submitted for the upcoming edition. Of the

7

total articles submitted,

5

were editorials. If she liked all the articles equally, and randomly selected

4

articles to go on the front cover, what is the probability that all of them are editorials? Write your answer as a decimal rounded to four decimal places. ____

Calculate Total Number of Ways: First, calculate the total number of ways to choose $4$ articles out of $7$ . This is a combination problem, so we use the formula for combinations: $C(n, k) = \frac{n!}{k!(n-k)!}$ . Calculate $C(7, 4)$ for the total number of ways to choose $4$ articles. $C(7, 4) = \frac{7!}{4!(7-4)!} = \frac{(7\cdot6\cdot5\cdot4!)}{(4!\cdot3\cdot2\cdot1)} = \frac{(7\cdot6\cdot5)}{(3\cdot2\cdot1)} = 35$ .
Calculate Ways to Choose Editorials: Next, calculate the number of ways to choose $4$ editorials out of the $5$ available. $\newline$ Use the combination formula again: $C(5, 4)$ . $\newline$ $C(5, 4) = \frac{5!}{(4!(5-4)!)} = \frac{(5\cdot4!)}{(4!\cdot1!)} = \frac{5}{1} = 5$ .
Find Probability: Now, find the probability by dividing the number of ways to choose $4$ editorials by the total number of ways to choose $4$ articles. $\newline$ Probability = Number of ways to choose $4$ editorials / Total number of ways to choose $4$ articles. $\newline$ Probability = $\frac{5}{35}$ .
Convert to Decimal: Simplify the fraction $\frac{5}{35}$ to get the decimal form. $\frac{5}{35} = \frac{1}{7}$ . Convert $\frac{1}{7}$ to a decimal and round to four decimal places. $\frac{1}{7} \approx 0.1429$ .

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