The number of clients received per month follows a Poisson distribution with a mean ($\lambda_{\text{month}}$) of 1. Because the months are independent, the number of clients received over a full year (12 months) follows a Poisson distribution where the mean is proportionally scaled: $\lambda_{\text{year}} = 1 \times 12 = 12$

The probability of exactly $k$ events in a Poisson distribution is given by the formula: $P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}$

For $k = 5$ and $\lambda = 12$: $P(X = 5) = \frac{e^{-12} \cdot 12^5}{5!} = \frac{0.000006144 \cdot 248832}{120} \approx 0.01274$

Therefore, the probability that the firm receives exactly 5 new clients in a year is approximately 0.01274.