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Jane likes to wear 2 bracelets, each a different color. If she has 8 bracelets each of a different color, how many combinations of 2 different-colored bracelets can she select to wear?
Combination problems involve choosing r combinations from n items. In this case, n = 8 bracelets and r = 2 bracelets
The formula for a combination of choosing
r unique ways from n possibilities is:
where n is the number of items and r is the unique arrangements.
What is n!, r!, (n - r)!?
n! signifies a factorial. n!, for example is shown in our
factorial lesson as being n! = n * (n - 1) * (n - 2) * .... * 2 * 1
Plugging in our factorial numbers, we get:
Calculate the numerator n!: n! = 8!
8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
8! = 40,320
Calculate the first denominator (n - r)!: (n - r)! = (8 - 2)!
(8 - 2)! = 6!
6! = 6 x 5 x 4 x 3 x 2 x 1
6! = 720
Calculate the second denominator r!: r! = 2!
2! = 2 x 1
2! = 2
Now calculate our combination value nCr for n = 8 and r = 2:8C
2 =
28Therefore, there are 28 unique ways to choose 2 different color bracelets from 8 total bracelets