First, we standardize the normal variable to find the probability that a single woman's middle finger is shorter than 8.2 cm:

\(P(X < 8.2) = P\left( Z < \frac{8.2 - 7.9}{0.44} \right) = P(Z < 0.6818)\)

Using the standard normal distribution table, \(P(Z < 0.6818) \approx 0.7524\).

Now, we use the binomial distribution to find the probability that exactly 3 out of 5 women have middle fingers shorter than 8.2 cm:

\(P(3) = \binom{5}{3} (0.7524)^3 (1 - 0.7524)^2\)

\(P(3) = 10 \times (0.7524)^3 \times (0.2476)^2\)

\(P(3) \approx 0.261\)