There are 25 suitcases, 5 of which are damaged. Three suitcases are selected at random. What is the probability that exactly 2 are damaged?

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∻Solution by Steps∻ ‖ step 1 ⋮ Calculate the number of ways to choose 2 damaged suitcases out of 5. ‖ ‖ step 2 ⋮ $$ \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 imes 4}{2 imes 1} = 10 $$ ‖ ‖ step 3 ⋮ Calculate the number of ways to choose 1 undamaged suitcase out of 20. ‖ ‖ step 4 ⋮ $$ \binom{20}{1} = \frac{20!}{1!(20-1)!} = \frac{20}{1} = 20 $$ ‖ ‖ step 5 ⋮ Calculate the total number of ways to choose 3 suitcases out of 25. ‖ ‖ step 6 ⋮ $$ \binom{25}{3} = \frac{25!}{3!(25-3)!} = \frac{25 imes 24 imes 23}{3 imes 2 imes 1} = 2300 $$ ‖ ‖ step 7 ⋮ Calculate the probability of choosing exactly 2 damaged suitcases. ‖ ‖ step 8 ⋮ $$ P( ext{2 damaged}) = \frac{\binom{5}{2} imes \binom{20}{1}}{\binom{25}{3}} = \frac{10 imes 20}{2300} = \frac{200}{2300} = \frac{2}{23} $$ ‖ ∻Answer∻ ⚹ $$ \frac{2}{23} $$ ⚹ ∻Key Concept∻ ⚹Combinations and Probability⚹ ∻Explanation∻ ⚹The probability of choosing exactly 2 damaged suitcases is found by dividing the number of ways to choose 2 damaged and 1 undamaged suitcase by the total number of ways to choose any 3 suitcases.⚹◊What is the formula for calculating the probability of selecting exactly 2 damaged suitcases out of 3 selected from a total of 25 suitcases?⍭ Generate me a similar question◊

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