A and B can complete a piece of work in $10$ days. B and C can do it in $15$ days. A and C can do it in $12$ days. A, B and C together can complete it in how many days?

Let the individual daily work efficiencies of $A,B,$ and $C$ be represented by $A,B,$ and $C$ respectively.
From the problem, we have:

$$A+B=\frac{1}{10}\text{— (Equation 1)}$$

$$B+C=\frac{1}{15}\text{— (Equation 2)}$$

$$A+C=\frac{1}{12}\text{— (Equation 3)}$$

Adding Equation 1, Equation 2, and Equation 3 together:

$$2(A+B+C)=\frac{1}{10}+\frac{1}{15}+\frac{1}{12}$$

Find the Least Common Multiple (LCM) of denominators $10,15,$ and $12$, which is $60$:

$$2(A+B+C)=\frac{6+4+5}{60}$$

$$2(A+B+C)=\frac{15}{60}$$

$$2(A+B+C)=\frac{1}{4}$$

Divide by $2$ to find the combined daily efficiency of $A,B,$ and $C$:

$$A+B+C=\frac{1}{4\times 2}=\frac{1}{8}$$

Since their collective work rate is $\frac{1}{8}$ of the work per day, they will take exactly $8$ days to complete the work together.