Answer
Three standing wave patterns.
Work Step by Step
The tension in the string is the weight of the mass on the end.
$$F_T=mg$$
We may find the speed of waves on the string.
$$v=\sqrt{\frac{F_T}{\mu}}=\sqrt{\frac{mg}{\mu}}$$
Next, we find the wavelength of the waves, using the given frequency of 60.0 Hz.
$$\lambda=\frac{v}{f}=\frac{1}{f}\sqrt{\frac{mg}{\mu}}$$
$$\lambda=\frac{v}{f}=\frac{1}{60.0Hz}\sqrt{\frac{(0.080kg)(9.80\;m/s^2)}{3.5\times10^{-4}\;kg/m}}=0.7888\;m$$
For there to be nodes at both ends and to form a standing wave, the length of the string must be an integer multiple of half a wavelength. The possible lengths are $\lambda/2$, $\lambda$, $3\lambda/2$, $2\lambda$...
This means the possibilities for lengths are 0.39m, 0.79m, 1.18m, 1.58m, etc.
If the length varies between 0.10m and 1.5m, only the first 3 solutions are valid.