To determine the total number of ways to pick combinations of dresses and shoes, we need to follow a step-by-step approach.
1. Calculate the number of ways to pick six dresses out of nine:
First, we compute the number of combinations of six dresses from a total of nine dresses. The combination formula is given by:
[tex]\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\][/tex]
where [tex]\( n \)[/tex] is the total number of items, and [tex]\( k \)[/tex] is the number of items to be chosen. Here, [tex]\( n = 9 \)[/tex] and [tex]\( k = 6 \)[/tex]:
[tex]\[
\binom{9}{6} = \frac{9!}{6!(9-6)!} = \frac{9!}{6! \cdot 3!}
\][/tex]
Performing this calculation, we get:
[tex]\[
\binom{9}{6} = 84
\][/tex]
2. Calculate the number of ways to pick two pairs of shoes out of five:
Next, we compute the number of combinations of two pairs of shoes from a total of five pairs. Using the same combination formula, where [tex]\( n = 5 \)[/tex] and [tex]\( k = 2 \)[/tex]:
[tex]\[
\binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5!}{2! \cdot 3!}
\][/tex]
Performing this calculation, we get:
[tex]\[
\binom{5}{2} = 10
\][/tex]
3. Calculate the total number of combinations:
The total number of ways to pick six dresses and two pairs of shoes is obtained by multiplying the combinations of the two separate events (dresses and shoes):
[tex]\[
\binom{9}{6} \times \binom{5}{2} = 84 \times 10 = 840
\][/tex]
So, the number of ways to pick six out of nine dresses followed by two out of five pairs of shoes is [tex]\( 840 \)[/tex]. Therefore, the correct answer is:
○ 840
