Given two 2's, "plus" can be changed to "times" without changing the result: 2+2=2·2. The solution with three numbers is easy too: 1+2+3=1·2·3. There are three answers for the five-number case. Which five numbers with this property has the largest sum?
1. Let the five numbers be \( a, b, c, d, e \). We need to find the numbers such that their sum equals their product:
   \[
   a + b + c + d + e = a \cdot b \cdot c \cdot d \cdot e
   \]

2. First, consider the case where all five numbers are equal. If all numbers are 2, then:
   \[
   2 + 2 + 2 + 2 + 2 = 10 \quad \text{and} \quad 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32
   \]
   Since \( 10 \neq 32 \), this is not a solution.

3. Next, consider the case where four of the numbers are 1. Let \( a = b = c = d = 1 \) and \( e \) be the fifth number:
   \[
   1 + 1 + 1 + 1 + e = 4 + e \quad \text{and} \quad 1 \cdot 1 \cdot 1 \cdot 1 \cdot e = e
   \]
   This gives:
   \[
   4 + e = e
   \]
   This is impossible since \( 4 + e \neq e \).

4. Now, consider the case where three of the numbers are 1. Let \( a = b = c = 1 \) and \( d, e \) be the other two numbers:
   \[
   1 + 1 + 1 + d + e = 3 + d + e \quad \text{and} \quad 1 \cdot 1 \cdot 1 \cdot d \cdot e = d \cdot e
   \]
   This gives:
   \[
   3 + d + e = d \cdot e
   \]
   Rearrange to find:
   \[
   d \cdot e - d - e = 3 \quad \text{or} \quad (d-1)(e-1) = 4
   \]
   The pairs \((d-1, e-1)\) that multiply to 4 are \((1, 4)\) and \((2, 2)\). Thus, the solutions are:
   \[
   (d, e) = (2, 5) \quad \text{and} \quad (d, e) = (3, 3)
   \]
   Therefore, the sets of numbers are \((1, 1, 1, 2, 5)\) and \((1, 1, 1, 3, 3)\).

5. Finally, consider the case where two of the numbers are 1. Let \( a = b = 1 \) and \( c, d, e \) be the other three numbers:
   \[
   1 + 1 + c + d + e = 2 + c + d + e \quad \text{and} \quad 1 \cdot 1 \cdot c \cdot d \cdot e = c \cdot d \cdot e
   \]
   This gives:
   \[
   2 + c + d + e = c \cdot d \cdot e
   \]
   We need to find integers \( c, d, e \) that satisfy this equation. Testing small values, we find:
   \[
   c = d = e = 2
   \]
   This gives:
   \[
   2 + 2 + 2 + 2 = 8 \quad \text{and} \quad 2 \cdot 2 \cdot 2 = 8
   \]
   Therefore, the set of numbers is \((1, 1, 2, 2, 2)\).

6. We have found three solutions:
   \[
   (1, 1, 1, 2, 5), \quad (1, 1, 1, 3, 3), \quad (1, 1, 2, 2, 2)
   \]
   The sums of these sets are:
   \[
   1 + 1 + 1 + 2 + 5 = 10, \quad 1 + 1 + 1 + 3 + 3 = 9, \quad 1 + 1 + 2 + 2 + 2 = 8
   \]

Conclusion:
The set of numbers with the largest sum is \((1, 1, 1, 2, 5)\) with a sum of \( \boxed{10} \).