What is the difference between the maximum value and the minimum value of the sum $a_1 + 2a_2 + 3a_3 + 4a_4 + 5a_5$ where $\{a_1,a_2,a_3,a_4,a_5\} = \{1,2,3,4,5\}$?

$ 
\textbf{(A)}\ 20
\qquad\textbf{(B)}\ 15
\qquad\textbf{(C)}\ 10
\qquad\textbf{(D)}\ 5
\qquad\textbf{(E)}\ 0
$
1. We are given the sum \( S = a_1 + 2a_2 + 3a_3 + 4a_4 + 5a_5 \) where \(\{a_1, a_2, a_3, a_4, a_5\} = \{1, 2, 3, 4, 5\}\). We need to find the difference between the maximum and minimum values of \( S \).

2. To find the maximum value of \( S \), we should assign the largest coefficients to the largest values of \(a_i\). Thus, we assign:
   \[
   a_5 = 5, \quad a_4 = 4, \quad a_3 = 3, \quad a_2 = 2, \quad a_1 = 1
   \]
   Substituting these values into \( S \), we get:
   \[
   S_{\text{max}} = 1 \cdot 1 + 2 \cdot 2 + 3 \cdot 3 + 4 \cdot 4 + 5 \cdot 5
   \]
   \[
   S_{\text{max}} = 1 + 4 + 9 + 16 + 25 = 55
   \]

3. To find the minimum value of \( S \), we should assign the smallest coefficients to the largest values of \(a_i\). Thus, we assign:
   \[
   a_1 = 5, \quad a_2 = 4, \quad a_3 = 3, \quad a_4 = 2, \quad a_5 = 1
   \]
   Substituting these values into \( S \), we get:
   \[
   S_{\text{min}} = 1 \cdot 5 + 2 \cdot 4 + 3 \cdot 3 + 4 \cdot 2 + 5 \cdot 1
   \]
   \[
   S_{\text{min}} = 5 + 8 + 9 + 8 + 5 = 35
   \]

4. The difference between the maximum and minimum values of \( S \) is:
   \[
   S_{\text{max}} - S_{\text{min}} = 55 - 35 = 20
   \]

The final answer is \(\boxed{20}\).