Let’s assume that, while working alone, A, B and C can individually finish the work in a, b and c days respectively.

∴ Part of work finished by A in one day = 1/a

∴ Part of work finished by B in one day = 1/b

∴ Part of work finished by C in one day = 1/c

A and B can do a piece of work in 10 days.

∴ Part of work finished by A and B in one day $(= \frac{1}{{\rm{a}}} + \frac{1}{{\rm{b}}} = \frac{1}{{10}})$ -----(i)

B and C can do a piece of work in 15 days.

∴ Part of work finished by B and C in one day $(= \frac{1}{{\rm{b}}} + \frac{1}{{\rm{c}}} = \frac{1}{{15}})$ -----(ii)

C and A can do a piece of work in 20 days.

∴ Part of work finished by C and A in one day $(= \frac{1}{{\rm{c}}} + \frac{1}{{\rm{a}}} = \frac{1}{{20}})$ -----(iii)

Adding equation (i), (ii) and (iii), we get:

$(2{\rm{\;}}\left( {\frac{1}{{\rm{a}}} + \frac{1}{{\rm{b}}} + \frac{1}{{\rm{c}}}} \right) = \frac{1}{{10}} + \frac{1}{{15}} + \frac{1}{{20}} = \frac{{13}}{{60}})$

$(\Rightarrow {\rm{\;}}\left( {\frac{1}{{\rm{a}}} + \frac{1}{{\rm{b}}} + \frac{1}{{\rm{c}}}} \right) = \frac{{13}}{{120}})$ -----(iv)

Subtracting equation (i) from equation (iv):

$(\frac{1}{{\rm{c}}} = \frac{{13}}{{120}} - \frac{1}{{10}} = \frac{{13 - 12}}{{120}} = \frac{1}{{120}})$