Answer:
105
Step-by-step explanation:
On the assumption that each switch must be a different type, we need to choose 1 type I switch from 2, 1 type II switch from 5 and 1 type III switch from 7.
The number of ways of picking [tex]r[/tex] objects from [tex]n[/tex] objects is [tex]\binom{n}{r}=\dfrac{n!}{r!(n-r)!}[/tex]
[tex]\binom{3}{1}=\dfrac{3!}{1!2!}=3[/tex]
[tex]\binom{5}{1}=\dfrac{5!}{1!4!}=5[/tex]
[tex]\binom{7}{1}= \dfrac{7!}{1!6!}=7 [/tex]
The total number of different choices = [tex]3\times5\times7=105[/tex]
If there's no restriction on the choices, we would need to pick 3 socks from a total of 15.
[tex]\binom{15}{3}=\dfrac{15!}{3!12!}=105[/tex]
