Suppose $11$ manufacturers produce a certain microchip, whose quality varies from manufacturer to manufacturer. If you were to select $7$ manufacturers at random, what is the chance that the selection would contain exactly $2$ of the best $3$?
Sample space: $11C7 = 330$ ways.
Number of events: $8C5 = 56$ ways.
$56/330$ or $0.1697$
Is my answer correct?
You are correct that there are $\binom{11}{7}$ ways to choose seven of the eleven manufacturers.
However, your count of the favorable cases is incomplete. If exactly two of the three best manufacturers are selected, then five of the other eight manufacturers must be selected. There are $\binom{3}{2}$ ways to select two of the three best manufacturers and $\binom{8}{5}$ ways to select five of the other eight manufacturers. Hence, the number of favorable cases is $$\binom{3}{2}\binom{8}{5}$$ Therefore, the probability that the selection contains exactly two of the best three manufacturers is $$\frac{\dbinom{3}{2}\dbinom{8}{5}}{\dbinom{11}{7}}$$