A sample of hydrogen atoms and in each atom , the electron is excited to the energy level n.What is the?maximum no. of unique wavelengths this sample can emit?please give a simple explanation. Its really confusing.Also i would like to know the difference between the above and below question.There is a hydrogen atom in the ground state . i t is excited to a higher energy level n .When it comes back to ground state, it emits radiation .What is the max. no. of unique wavelength it can emit?

As it seems to me both questions are one & same.You see whenever an electron is in  higher energy level it loses exact quanta of energy in the form of radiation emitted to go in the lower energy level.For hydrogen atom this energy is inversely proportional to the square of level that electron occupy.

Now from  level 'n' maximum transitions could be 'n-1' as electron could end up in any of lower 'n-1' levels.As you could verify that each transition is unique for the value of enery released as radiation.So you've 'n-1' unique wavelenths corresponding to them.

Similarly for electron in 'n-1'th level there are 'n-2' unique wavelengths & so on upto 1 for 'n=1' (one above ground level).

Adding all give max. no. of unique wavelengths=n*(n-1)/2   {or,ⁿC₂ ;i.e. choosing 2 levels out of n}

For 2nd version it may be explicitly said that no in-between transitions are taken into consideration,only transition to ground state is counted then you've distinct 'n-1' wavelengths,otherwise both Q's are same statements.

Here you must know that for the 1st version of the Q i neglect any interaction between radiation emitted by one atom  & another atom.I presumed them to be single entities in behaviour,which couldn't be so.Because in a sample of hydrogen atoms series of radiation attacking some electrons may propel them to higher energy states than 'n' which in turn cause emission of more energetic radiations when these electrons fall back to lower levels which in turn excite more & more like an avalanche.You see it's impossible then to determine exact no. of possible transitions !!!!

Above argument is quite valid for low pressure & high temp. sample with infinite volume to occupy.

I hope you get this.Any further query in this is answered fully just make your Q to the point.(what u wanna know!)