Question

You are planning a trip and need to pack for 7 days. You have 10 outfits to choose from. You are trying to decide how many different ways you could choose 7 outfits from the 10 you have. Would order matter? Would you choose Permutations or Combinations? How many ways are there?

Answer 1

In this situation, you are trying to choose 7 outfits from a total of 10 outfits. Since the order in which you select the outfits matters (as you would likely wear a different outfit on each day), you are dealing with permutations rather than combinations.

To find the number of permutations of 10 outfits taken 7 at a time, you can use the formula for permutations:

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

Where \( n \) is the total number of items to choose from (in this case, 10 outfits), and \( r \) is the number of items to choose (in this case, 7 outfits).

Plugging in the values:

\[
P(10, 7) = \frac{10!}{(10-7)!} = \frac{10!}{3!}
\]

Now, calculating \( 10! \) (the factorial of 10) and \( 3! \) (the factorial of 3):

- \( 10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \)
- \( 3! = 3 \times 2 \times 1 = 6 \)

Thus:

\[
P(10, 7) = \frac{10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4}{3 \times 2 \times 1} = \frac{10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4}{6}
\]

Calculating the numerator:

\[
10 \times 9 = 90
\]
\[
90 \times 8 = 720
\]
\[
720 \times 7 = 5040
\]
\[
5040 \times 6 = 30240
\]
\[
30240 \times 5 = 151200
\]
\[
151200 \times 4 = 604800
\]

Finally, divide by 6:

\[
\frac{604800}{6} = 100800
\]

Therefore, the number of different ways to choose 7 outfits from 10, where the order matters, is **100800**.