Without direct evaluation- determine which of the following two quantities is larger:
(24!)2 vs 2424
I'd estimate the common log of 24 to be about 1.4, and 24 times that to be about 34, so the RHS would be around a 34-digit number.
Then the size comparison for 24! would be about a 17-digit number (the square root of a 34-digit number), to exceed or fall short.
There are 15 factors of 10 or higher that go into 24!, and below that, 2*5, etc. add more digits, and the 15 factors previously mentions, 14 are definitely larger than 10.
I'd say the factorial squared not only is larger, but by orders of magnitude larger.
Computer check:
>> factorial(24)^2
ans =
3.84956219213331e+47
>> 24^24
ans =
1.33373577685028e+33
An alternative way of arriving at the answer without direct evaluation of the given comparison is to see that early on, even at n=3, (n!)^2 exceeds n^n:
>> for i= 1:10 fprintf('%2d %14d %14d\n',[i,factorial(i)^2, i^i])
end
1 1 1
2 4 4
3 36 27
4 576 256
5 14400 3125
6 518400 46656
7 25401600 823543
8 1625702400 16777216
9 131681894400 387420489
10 13168189440000 10000000000
>>
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Posted by Charlie
on 2024-10-03 09:19:32 |
solution
