Try this beautiful problem from the Pre-RMO, 2017 based on Ordered triples.
What is the number of triples (a,b,c) of positive integers such that abc=108?
Largest number of triples
Combinatrics
Integers
Answer: is 60.
PRMO, 2017, Question 21
Elementary Number Theory by David Burton
abc=\(3^{3}2^{2}\)
a=\(3^{\alpha_1}2^{\beta_1}\), b=\(3^{\alpha_2}2^{\beta_2}\), c=\(3^{\alpha_3}2^{\beta_3}\)
\({\alpha_1}+{\alpha_2}+{\alpha_3}=3\), \({\beta_1}+{\beta_2}+{\beta_3}=2\)
\({5 \choose 2}\), \({4 \choose 2}\)
total= \({5 \choose 2} \times {4 \choose 2}\)=(10)(6)=60 ways.
