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.
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