(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 70 colors. How many mini-boxes are possible?
Question
(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 70 colors. How many mini-boxes are possible?
Solution
The problem can be solved using the concept of combinations in mathematics.
A combination is a selection of items where order does not matter. In this case, the order in which the colors are selected does not matter, only which colors are selected.
The formula for combinations is:
C(n, r) = n! / [r!(n-r)!]
where:
- n is the total number of options,
- r is the number of options chosen at a time,
- "!" denotes a factorial, meaning the product of all positive integers up to that number.
In this case, n = 70 (the total number of colors) and r = 5 (the number of colors to be chosen for the mini-box).
So, the number of possible mini-boxes is:
C(70, 5) = 70! / [5!(70-5)!]
Calculating this gives the total number of possible mini-boxes.
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