We know that: \(\displaystyle{n}{\left({X}\cdot{Y}\right)}={n}{\left({X}\right)}\cdot{n}{\left({Y}\right)}\), \(\displaystyle{n}{\left({X}\cdot{Z}\right)}={n}{\left({X}\right)}\cdot{n}{\left({Z}\right)}\), \(\displaystyle{n}{\left({Y}\cdot{Z}\right)}={n}{\left({Y}\right)}\cdot{n}{\left({Z}\right)}\),

and

\(\displaystyle{x}{\left({X}\cdot{Y}\cdot{Z}\right)}={n}{\left({X}\right)}\cdot{n}{\left({Y}\right)}\cdot{n}{\left({Z}\right)}\)

Therefore, \(\displaystyle{n}{\left({X}\right)}\cdot{n}{\left({Y}\right)}={24},{n}{\left({X}\right)}\cdot{n}{\left({Z}\right)}={15}\)

and

\(\displaystyle{n}{\left({Y}\right)}\cdot{n}{\left({Z}\right)}={40}\)

Multiplying the three equations, we get: \(\displaystyle{\left({n}{\left({X}\right)}\right)}^{{2}}{\left({n}{\left({Y}\right)}\right)}^{{2}}{\left({n}{\left({Z}\right)}\right)}^{{2}}={24}\cdot{15}\cdot{40}={14400}\)

Thus, \(\displaystyle{\left({n}{\left({X}\right)}\cdot{n}{\left({Y}\right)}\cdot{n}{\left({Z}\right)}\right)}^{{2}}={14400}\to{n}{\left({X}\right)}\cdot{n}{\left({Y}\right)}\cdot{n}{\left({Z}\right)}={120}\)

Finally, using (*), \(\displaystyle{n}{\left({X}\cdot{Y}\cdot{Z}\right)}={120}\)