Discussion
Three particles are to be distributed in four non-degenerate energy levels. The possible number of ways of distribution: (i) for distinguishable particles, and (ii) for identical Bosons, respectively, is
Answer: C
The counts are (i) 64 and (ii) 20.
(i) Distinguishable particles: each of the 3 particles independently chooses one of 4 levels, so there are \(4^{3} = 64\) arrangements.
(ii) Identical bosons: only the occupation numbers matter, and any number may share a level. The number of ways to put 3 identical objects into 4 boxes is
\[ \binom{3 + 4 - 1}{3} = \binom{6}{3} = 20 \]
24 is \(4 \times 3 \times 2\), the count for distinguishable particles that may not share a level. 4 is \(\binom{4}{3}\), the count for fermions (at most one per level). 16 has no basis.
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