Mathematica competitions in Croatia

Denote the number of arrangements of $n$ envelopes as $K_n$. Assume we are given an arrangement of $n$ envelopes. If we remove the smallest envelope, we obtain a possible arrangement of $n-1$ remaining envelopes. On the other hand, if we are given a possible arrangement of the remaining $n-1$ envelopes, we can put the smallest one directly in the biggest one, or in any of the remaining $n-2$ envelopes. Therefore, $K_n$ is exactly $n-1$ times larger than $K_{n-1}$. We conclude that $K_n=(n-1)\cdot K_{n-1}=\cdots =(n-1)\cdot (n-2)\cdots 2\cdot K_2$. For $n=2$, the only possible arrangement is putting the smaller envelope inside the larger one, so $K_2=1$. Therefore $K_n=(n-1)\cdot (n-2)\cdots 1$. For 100 envelopes, the number of different arrangements is $$K_{100}=99\cdot 98\cdots 1=99!$$