To solve the problem of finding the number of vehicles remaining in the bin after the store sells [tex]\(\frac{1}{3}\)[/tex] of the vehicles, we first need to determine the total number of vehicles in the bin and then compute how many vehicles are sold and how many remain.
1. Determine the total number of vehicles:
The bin contains [tex]\(b\)[/tex] bikes and [tex]\(c\)[/tex] cars. Thus, the total number of vehicles in the bin is [tex]\(b + c\)[/tex].
2. Calculate the number of vehicles sold:
The store sells [tex]\(\frac{1}{3}\)[/tex] of the total number of vehicles. Therefore, the number of vehicles sold is:
[tex]\[
\frac{1}{3} \times (b + c)
\][/tex]
3. Find the number of vehicles remaining:
To find the remaining vehicles after the sale, we subtract the number of vehicles sold from the total number of vehicles:
[tex]\[
\text{Remaining vehicles} = (b + c) - \frac{1}{3} \times (b + c)
\][/tex]
4. Simplify the expression:
We can factor out [tex]\(b + c\)[/tex] from the expression:
[tex]\[
(b + c) - \frac{1}{3} (b + c) = \left(1 - \frac{1}{3}\right) (b + c)
\][/tex]
Simplify [tex]\(1 - \frac{1}{3}\)[/tex]:
[tex]\[
1 - \frac{1}{3} = \frac{3}{3} - \frac{1}{3} = \frac{2}{3}
\][/tex]
Therefore, the expression for the remaining number of vehicles is:
[tex]\[
\frac{2}{3} (b + c)
\][/tex]
Thus, the correct answer is:
C. [tex]\(\frac{2}{3}(b+c)\)[/tex]
