Question

Kimberly has seven colors of lanyards. She uses three different colors to make a key chain. How many different combinations can she choose?
(1 Point)


6

35

210

5,040

Answer 1

To determine how many different combinations of three colors Kimberly can choose from seven available colors of lanyards, we need to use the combination formula, which is given by:

\[
C(n, r) = \frac{n!}{r!(n - r)!}
\]

where:
- \( n \) is the total number of items to choose from (in this case, 7 colors),
- \( r \) is the number of items to choose (in this case, 3 colors),
- \( ! \) denotes factorial, which is the product of all positive integers up to that number.

Plugging in the values:

\[
C(7, 3) = \frac{7!}{3!(7 - 3)!} = \frac{7!}{3! \cdot 4!}
\]

Calculating the factorials:

- \( 7! = 7 \times 6 \times 5 \times 4! \) (we can stop at \( 4! \) since it cancels out)
- \( 3! = 3 \times 2 \times 1 = 6 \)
- \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)

Now substituting back into the combination formula:

\[
C(7, 3) = \frac{7 \times 6 \times 5 \times 4!}{3! \cdot 4!} = \frac{7 \times 6 \times 5}{3!} = \frac{7 \times 6 \times 5}{6}
\]

We can simplify this:

\[
= 7 \times 6 \times 5 / 6 = 7 \times 5 = 35
\]

Thus, the number of different combinations of three colors that Kimberly can choose from seven colors of lanyards is **35**.