X different wavelength may be observed in the spectrum from a...

Use the concept of hydrogen like atom spectrum when the atoms are excited to a sixth excited state, to find different wavelengths observed in the spectrum.

The total possible transitions are,

$\dfrac{n\left(n - 1\right)}{2} = \dfrac{6 \times 5}{2} = 15$

Here $n$ is the principal quantum number.