Spectral Lines From Energy Levels

When electrons de-excite from a level with principal quantum number $n$, the maximum number of distinct spectral lines produced by all possible downward transitions is $\frac{n(n-1)}{2}$, because each line corresponds to a unique pair of levels between which a transition can occur. For $n=4$ this gives $\frac{4\times 3}{2}=6$ lines. The value 4 wrongly counts only transitions ending at the ground state ($4\to 1,3\to 1,2\to 1$ plus one extra) and misses intermediate jumps such as $4\to 3$, $4\to 2$, and $3\to 2$. The value 3 counts only the direct drops from $n=4$ to each lower level. The value 10 corresponds to $n=5$ using the same formula and overcounts. This is the standard NCERT combinatorial result for hydrogen emission. A plausibility check by listing the pairs ($4\to 3,4\to 2,4\to 1,3\to 2,3\to 1,2\to 1$) confirms exactly six distinct transitions, matching the formula.