A gaseous mixture contains hydrogen atoms in the 4th excited state and helium ion; spectral lines emitted when all these return to the ground state


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> A gaseous mixture contains hydrogen atoms in the 4th excited state, $\ce{He^+}$ ions in 3d excited state, and $\ce{Li^{2+}}$ in 2nd excited state. Find the number of **distinct** spectral lines emitted when all these ions return to the ground state.
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Number of lines emitted $=\displaystyle\frac{n(n-1)}{2}$, where $n$ is the number of the shell the electron is in.


If we simply add the number of lines, without considering repetitions, we get $10+6+3=19$ lines.


However since there are 3 atoms, there will be some cases of repetition. Now we can always find out the repetitive cases by listing down the energy of each transition, but that will be very long, and there is potential for error since there are 19 lines. Is there a better method to solve this question.


Energy of transition $=\displaystyle13.6z^2\left|\frac{1}{n\_1^2}-\frac{1}{n\_2^2}\right|\ \mathrm{eV}$


