If 10 men can complete a piece of work in 15 days, and 15 women can complete the same work in 12 days, in how many days will 10 men and 15 women working together complete the work?

10 men can complete the work in 15 days.
So, in 1 day, 10 men complete $\frac{1}{15}$ of the work.

15 women can complete the work in 12 days.
So, in 1 day, 15 women complete $\frac{1}{12}$ of the work.

When working together, their daily work rates add up.
Combined daily work rate $= \frac{1}{15} + \frac{1}{12}$

The least common multiple (LCM) of 15 and 12 is 60.

So, 10 men and 15 women together complete $\frac{3}{20}$ of the work in 1 day.

If they complete $\frac{3}{20}$ of the work in 1 day, then the total time to complete the work is the reciprocal of their daily work rate.
Time taken $= \frac{1}{\frac{3}{20}} = \frac{20}{3}$ days.

Comparing this with the given options:

a) $6 \frac{2}{3}$ days

b) $8 \frac{1}{3}$ days

c) $7 \frac{2}{3}$ days

d) None of these

The calculated answer matches option a).

The final answer is $\boxed{\text{a) } 6 \frac{2}{3} \text{ days}}$.