Question

fourteen musical acts have asked to perform at springfield high schools winter talent show. eight of them play acoustic music and 6 play electric music. unfortunately, there is only room on the program for 10 acts - the others will have to wait until the spring show.
(a) how many different groups of 10 acts can be selected to play in the winter show?
(b) if the 10 musical acts for the winter show are randomly selected, what is the probability that all of the electric acts get to play in the show?
(c) if the 10 musical acts for the winter show are randomly selected, what is the probability that 5 acoustic acts and 5 electric acts get to play in the show?
(round to 3 decimal places. leave your answer in decimal form.)
(round to 3 decimal places. leave your answer in decimal form.)

Explanation:

Step1: Calculate combinations for part (a)

Use combination formula \(C(n,k)=\frac{n!}{k!(n - k)!}\), where \(n = 14\), \(k=10\).

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Step2: Calculate probability for part (b)

Number of ways to choose 10 acts with all 6 electric: Choose 6 electric (only 1 way as there are 6 electric acts) and 4 acoustic (\(C(8,4)\)).

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Probability \(P=\frac{C(8,4)}{C(14,10)}=\frac{70}{1001}\approx0.070\)

Step3: Calculate probability for part (c)

Number of ways to choose 5 electric (\(C(6,5)\)) and 5 acoustic (\(C(8,5)\)).

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Number of favorable cases \(=6\times56 = 336\)
Probability \(P=\frac{336}{1001}\approx0.336\)

Answer:

(a) \(1001\)
(b) \(0.070\)
(c) \(0.336\)