total permutation and factorial n!
P(n,n) = n! counts every possible order of n distinct elements, with none left out: it is the special case of permutation where r equals n. For 10 elements, 10! = 3,628,800 different orders; for 20, the total already passes 2.4 quintillion.
Orders only 3 of the 10 runners, leaving 7 out; it is this tool's 'no repetition' mode with r smaller than n, different from the total permutation.
That is 26 exact digits, well below the 10¹⁰⁰ ceiling: the result shows in full, without switching to the limit notation.
69! still shows all 99 digits in full; 70! is the first factorial to cross the 10¹⁰⁰ ceiling and trigger the limit message.
A permutation is an ordered arrangement of r elements chosen from a set of n; order matters, so AB and BA count as different permutations. Without repetition the number of arrangements is P(n, r) = n! / (n − r)!. For example, arranging 3 of the letters A, B, C, D gives P(4, 3) = 4 · 3 · 2 = 24 ordered sequences.
An arrangement is P(n,r) with r smaller than n: it orders only part of the elements, like 3 runners out of 10 on a podium (720 ways). A total permutation is the case r = n, when every element enters the ordering with none left out, like the full set of 10 runners (10! = 3,628,800 ways).
There is exactly one way to order zero elements, the empty sequence. The relation n! = n × (n-1)! only holds for n = 1 if 0! equals 1, since 1! = 1 × 0! = 1.
Up to 69, because 69! still stays below the 10¹⁰⁰ display ceiling, with 99 full digits. From 70! onward, the value crosses that ceiling and the tool shows '> 10¹⁰⁰' instead of the whole number, even though the internal BigInt calculation stays exact.
P(n, r) = n! / (n−r)!
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