How many moles are equal to 3.67X10^23 formula units of potassium permanganate

first let's calculate the energy involved in the transistions using planck's formula e = hf. with given wavelength, we can replace f by c/lambda. hence the formula will then become e = hc/lambda. energy units will be in ev (electron volts). just divide the energy units in j by electron charge to convert to ev.
(1) e1 = 108.79 ev
(2) e2 = 22.95 ev
(3) e3 = 17.00 ev

since there are no external electric and magnetic fields, then we can ignore stark and zeeman effects respectively. so, we only need to consider the n quantum number.
the energy of an electron in a nucleus of z charge and at nth orbit is given by.
e_n = -13.6*z^2/n^2 ev
let's calculate the energies of the electrons at different energy levels. for helium, z = 2. using the four lowest levels,
e_1 = -54.40 ev
e_2 = -13.60 ev
e_3 = -6.04 ev
e_4 = -3.40 ev
this time we need to calculate the transistion energies for every two pairs of energy levels. there are 4c2 = 6 possible transistion energies.
e_14 = e_4 - e_1 = 51.00 ev
e_13 = e_3 - e_1 = 48.36 ev
e_12 = e_2 - e_1 = 40.80 ev
e_24 = e_4 - e_2 = 10.20 ev
e_23 = e_3 - e_2 = 7.56 ev
e_34 = e_4 - e_3 = 2.64 ev

so it seems that no energy transition matches the change in energy for each jump. so the given wavelengths must be wrong.