Suppose we want to choose 6 colors, without replacement, from 11 distinct colors.(If necessary, consult a list of formulas.)(a) How many ways can this be done, if the order of the choices matters?(b) How many ways can this be done, if the order of the choices does not matter?
Question
Suppose we want to choose 6 colors, without replacement, from 11 distinct colors.(If necessary, consult a list of formulas.)(a) How many ways can this be done, if the order of the choices matters?(b) How many ways can this be done, if the order of the choices does not matter?
Solution
(a) If the order of the choices matters, we use the permutation formula. The number of ways to choose 6 colors from 11, without replacement and where order matters, is given by the formula for permutations of n items taken r at a time, which is nPr = n! / (n-r)!. Here, n is the total number of items (11 colors), and r is the number of items to choose (6 colors). So, the calculation is 11P6 = 11! / (11-6)! = 1110987*6 = 332,640 ways.
(b) If the order of the choices does not matter, we use the combination formula. The number of ways to choose 6 colors from 11, without replacement and where order does not matter, is given by the formula for combinations of n items taken r at a time, which is nCr = n! / [r!(n-r)!]. Here, n is the total number of items (11 colors), and r is the number of items to choose (6 colors). So, the calculation is 11C6 = 11! / [6!(11-6)!] = 11109876 / (654321) = 462 ways.
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