Answer the questions below.(If necessary, consult a list of formulas.)(a) A company that makes crayons is trying to decide which 5 colors to include in a promotional mini-box of 5 crayons. The company can choose the 5 mini-box colors from its collection of 75 colors. How many mini-boxes are possible?(b) How many different committees of size 3 can be formed from 14 people?
Question
Answer the questions below.(If necessary, consult a list of formulas.)(a) A company that makes crayons is trying to decide which 5 colors to include in a promotional mini-box of 5 crayons. The company can choose the 5 mini-box colors from its collection of 75 colors. How many mini-boxes are possible?(b) How many different committees of size 3 can be formed from 14 people?
Solution
(a) The number of ways to choose 5 colors from a collection of 75 is given by the combination formula, which is C(n, r) = n! / [(n-r)! * r!], where n is the total number of items, r is the number of items to choose, and "!" denotes factorial.
In this case, n = 75 (the total number of colors) and r = 5 (the number of colors to choose for the mini-box).
So, the number of possible mini-boxes is C(75, 5) = 75! / [(75-5)! * 5!] = 17,259,390.
(b) The number of ways to form a committee of 3 people from a group of 14 is also given by the combination formula.
In this case, n = 14 (the total number of people) and r = 3 (the number of people to choose for the committee).
So, the number of possible committees is C(14, 3) = 14! / [(14-3)! * 3!] = 364.
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