According to the PMBOK® Guide, specifically within the Estimate Activity Durations process, project managers use Three-Point Estimating to improve the accuracy of activity duration estimates. This technique considers uncertainty and risk by using three estimates:

Optimistic ($t_O$): The best-case scenario (10 days).

Most Likely ($t_M$): The most realistic scenario (15 days).

Pessimistic ($t_P$): The worst-case scenario (20 days).

There are two common formulas used for three-point estimating. The question specifically asks for the Triangular Distribution:

The Formula:

$$E = \frac{t_O + t_M + t_P}{3}$$

The Calculation:

$$E = \frac{10 + 15 + 20}{3}$$

$$E = \frac{45}{3}$$

$$E = 15 \text{ days}$$

Why other options are incorrect:

Option A (10 days): This is simply the Optimistic estimate ($t_O$), which ignores the most likely and pessimistic scenarios.

Option C (12.5 days): This value does not correspond to any standard PMBOK duration estimation formula based on the numbers provided.

Option D (5 days): This is significantly lower than even the optimistic estimate and has no mathematical basis in this context.

Note on Beta Distribution (PERT):

It is important to distinguish this from the Beta Distribution (often used in PERT), which gives more weight to the " Most Likely " estimate. If the question had asked for the Beta distribution, the calculation would be:

$$E = \frac{t_O + 4t_M + t_P}{6} = \frac{10 + (4 \times 15) + 20}{6} = \frac{90}{6} = 15 \text{ days}$$