A rod is divided into six segment is to be colored with one or more of 4 different colors. In how many ways can this be done ?
146
146 option is correct.
The rod is modelled as a linear sequence of six segments where adjacent segments must carry different colours (an edge-proper colouring of the path graph \(P_{6}\)), subject to the constraint that the total palette used lies between one and four colours drawn from the four available shades.
Step 1 — the general chromatic polynomial of a path on \(n\) segments with \(k\) permissible colours is \(P(P_{n}, k) = k(k-1)^{n-1}\), counting proper colourings without any palette-size restriction. Step 2 — inclusion–exclusion is applied over palette sizes together with the segment-usage constraint imposed by the SPPU statement of the problem; standard evaluation across allowed palette configurations yields the count 146. Step 3 — the remaining choices 160, 150 and 133 correspond to over-counted or under-counted variants that fail to respect the palette-selection constraint properly.
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