Suppose we want to choose 3 colors, without replacement, from the 4 colors red, blue, green, and purple.(If necessary, consult a list of formulas.)(a) How many ways can this be done, if the order of the choices is relevant?(b) How many ways can this be done, if the order of the choices is not relevant?
Question
Suppose we want to choose 3 colors, without replacement, from the 4 colors red, blue, green, and purple.(If necessary, consult a list of formulas.)(a) How many ways can this be done, if the order of the choices is relevant?(b) How many ways can this be done, if the order of the choices is not relevant?
Solution
(a) If the order of the choices is relevant, we use the permutation formula. The number of ways to choose 3 colors from 4, when order matters, is given by P(n, r) = n! / (n-r)!. Here, n is the total number of options (4 colors), and r is the number of options chosen (3 colors). So, P(4, 3) = 4! / (4-3)! = 24.
(b) If the order of the choices is not relevant, we use the combination formula. The number of ways to choose 3 colors from 4, when order does not matter, is given by C(n, r) = n! / [r!(n-r)!]. Here, n is the total number of options (4 colors), and r is the number of options chosen (3 colors). So, C(4, 3) = 4! / [3!(4-3)!] = 4.
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