Assume that a hydrogen atom's electron has been excited to the n 5 level. how many different wavelengths of light can be emitted as this excited atom loses energy?

From n = 5, the electron can drop to n = 4, 3, 2, 1 . Thus, there will be 4 possibilities

Now, the electron that dropped to n = 4 can drop to n = 3, 2, 1. This will give 3 possibilities

The electron at n = 3, can drop to n = 2, 1. This gives 2 possibilities

Finally, the electron at n = 2 can drop to n=1. This gives 1 possibility

Thus in total the number of ways in which the electron can de-excite is :

4+3+2+1 = 10

10 different wavelengths of light can be emitted as the electron loses energy from n = 5 level