## Task A-2.5.

One hundred square envelopes of different sizes are arranged so that for any two different envelopes, the smaller envelope is inside the larger one or the envelopes are separate from each other. At the same time, both the smaller and the larger envelopes can contain other envelopes. Two arrangements are considered different if there are two envelopes that are one inside the other in one arrangement, but not in the other. How many different arrangements are there in which all other envelopes are inside the largest envelope?
## First Solution.

Consider the same problem with a total of $n$ envelopes and let $K_{n}$ denote the desired number of arrangements.

Given one arrangement of $n$ envelopes, if we remove the smallest envelope, we are left with an arrangement of the remaining $n-1$ envelopes. On the other hand, if we are given one possible arrangement of the remaining $n-1$ envelopes, the smallest envelope can be placed inside the largest envelope or in any of the remaining $n-2$ envelopes. Therefore, $K_{n}$ is exactly $n-1$ times larger than $K_{n-1}$.

Thus, $K_{n}=(n-1) \cdot K_{n-1}=(n-1) \cdot(n-2) \cdot K_{n-2}=\cdots=(n-1) \cdot(n-2) \cdots 2 \cdot K_{2}$.

When we have only two envelopes, the only possible way is to place the smaller one inside the larger one, so $K_{2}=1$. Therefore, the desired number of arrangements is $K_{n}=(n-1) \cdot(n-2) \cdots 1$.

For one hundred envelopes, the number of different arrangements is $K_{100}=99 \cdot 98 \cdots 1$, i.e., the product of the first 99 natural numbers.