5. Determine the number of ways to represent a natural number $n$ as the sum of several (two or more) natural numbers, where the order matters. (For example, for $n=4$, we have the following possibilities: $3+1, 2+2, 1+3, 2+1+1$, $1+2+1, 1+1+2, 1+1+1+1$, i.e., a total of 7 desired ways.)

## Fourth grade - B category
5. Let's consider $n$ objects (e.g., balls) arranged in a row. Each representation in the problem corresponds to placing barriers between some of the balls in such a way that the length of the $i$-th consecutive block of balls without a barrier represents exactly the $i$-th addend in the observed sum. (For example, for $n=11$, the representation $11=1+5+3+2$ corresponds to the following configuration of balls and barriers: $|\bullet \bullet \bullet \bullet| \bullet \bullet \bullet \mid \bullet \bullet$.) Clearly, this assignment is a bijection, so it is sufficient to count how many ways we can place the barriers between the balls. We have $n-1$ positions where we can place barriers (between every two consecutive balls), and for each of them, we choose whether to place a barrier or not; from the total number of ways obtained, we subtract 1, since we do not count the possibility of placing no barriers at all (since we must have at least two addends, according to the problem). Therefore, the answer is $2^{n-1}-1$.

## Fourth Grade - B Category