 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2005/paper-1/7/b/iii/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

For the stated

N_{A} = 6.022137 \times 1 0^{23}

, the leading term

N_{A} (ln N_{A} - 1)

dominates the logarithmic correction

a ln N_{A} + b

. Thus

ln (N_{A}!) \approx L (N_{A}) ≃ 3.237194 \times 1 0^{25} .

This uses the numerical value specified in the question; such a large logarithm can be computed without constructing the factorial itself.

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