Challenge: Combinations vs Permutations

Swap the loop order: amount in outer loop, coins in inner loop. Now you count permutations instead of combinations. For amount $3$ with coins $[1, 2]$: combinations are $[1,1,1]$, $[1,2]$ ($2$ ways).

Permutations are $[1,1,1]$, $[1,2]$, $[2,1]$ ($3$ ways). Why does loop order matter? Coins-first keeps you never use a smaller coin after a larger one (avoiding $[2,1]$ after counting $[1,2]$). Amount-first allows any order. Try both orders on coins $[1, 2, 5]$ and amount $5$. You should get $4$ combinations but more permutations.