See the wiki page
https://en.wikipedia.org/wiki/Factorial\_number\_system
for information on the Factorial Number System.
First, we write the base-10 number 60 in the Factorial
number system. We calculate
$60 \div 1 = 60$, remainder 0
$60 \div 2 = 30$, remainder 0
$30 \div 3 = 10$, remainder 0
$10 \div 4 = 2$, remainder 2
$2 \div 5 = 0$, remainder 2
This gives $60_{10} = 2:2:0:0:0_!$
Thus, the 60th permutation of 1, 2, 3, 4, 5 in lexicographic order
is $23145$.
