- The number of ways to distribute \( N_0 \) particles into \( N \) slots is given by the binomial coefficient \( \binom{N}{N_0} \). Since \( N_0 + N_1 = N \), this is also \( \binom{N}{N_1} \).
- Using Stirling's approximation, \( \ln(N!) \approx N\ln(N) - N \),
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