Exercise 2: Estimating O(n!) Running Time at n=15 — Possible Solution
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GIVEN
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1 billion (1e9) operations per second, n = 15

STEP 1: TOTAL OPERATIONS
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15! = 1,307,674,368,000 (~1.31 x 10^12 operations)

STEP 2: CONVERTING TO TIME
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seconds = 1.31 x 10^12 / 1 x 10^9 = 1,307.7 seconds

STEP 3: CONVERTING TO A HUMAN-READABLE UNIT
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1,307.7 seconds / 60 = 21.79 minutes

RESULT
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An O(n!) algorithm at n=15 would take approximately 21.8 minutes at
1 billion operations per second.

COMPARING TO THE O(2^n) ESTIMATE
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This exercise's operation count (~1.31 x 10^12 at n=15) is almost
identical in magnitude to Exercise 1's O(2^n) operation count
(~1.10 x 10^12 at n=40) - roughly the same total amount of work. But
15! reached that magnitude at n=15, while 2^40 needed n=40 to reach a
comparable size - factorial growth needed less than half the input
size to produce roughly the same total operation count. This confirms
factorial growth is dramatically faster than exponential growth: per
this chapter's own tables, n! overtakes 2^n by a wide and rapidly
growing margin as n increases further past this point.

WHY THIS WORKS AS AN ANSWER
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The total operation count and time estimate are computed using the
exact same method as Exercise 1, and the comparison between the two
algorithms is made concrete by noting how much SMALLER an n was needed
for the factorial case to reach a similar total operation count,
rather than just asserting factorial growth is "faster" in the
abstract.