There are finite many coins in David’s purse. The values of these coins are pair wisely distinct positive integers. Is that possible to make such a purse, such that David has exactly $2020$ different ways to select the coins in his purse and the sum of these selected coins is $2020$?
To determine if it is possible to construct a purse with coins such that there are exactly 2020 different ways to select the coins and the sum of these selected coins is 2020, we can follow these steps:

1. **Identify the Subset Sum Problem:**
   We need to find subsets of coins whose sums equal 2020. The number of such subsets should be exactly 2020.

2. **Construct a Simple Example:**
   Consider the set \( S = \{1, 2, 4, 756, \dots, 1264\} \). We need to verify if this set can provide the required number of subsets.

3. **Subsets of Small Numbers:**
   The numbers \( \{1, 2, 4\} \) can form subsets with sums ranging from 0 to 7:
   \[
   \begin{aligned}
   &\{ \} \rightarrow 0, \\
   &\{1\} \rightarrow 1, \\
   &\{2\} \rightarrow 2, \\
   &\{4\} \rightarrow 4, \\
   &\{1, 2\} \rightarrow 3, \\
   &\{1, 4\} \rightarrow 5, \\
   &\{2, 4\} \rightarrow 6, \\
   &\{1, 2, 4\} \rightarrow 7.
   \end{aligned}
   \]
   Thus, all possible sums from 0 to 7 can be achieved using these three coins.

4. **Pairs of Larger Numbers:**
   We need to find pairs of numbers from the range \([756, 1264]\) that add up to sums from 2013 to 2020. Let's denote these pairs as \((a, b)\) where \(a + b = 2020 - k\) for \(k = 0, 1, \ldots, 7\).

5. **Counting the Pairs:**
   For each \(k\), we need to count the number of pairs \((a, b)\) such that \(a + b = 2020 - k\). The number of such pairs is:
   \[
   \begin{aligned}
   &\text{For } k = 0: \quad 254 \text{ pairs}, \\
   &\text{For } k = 1: \quad 254 \text{ pairs}, \\
   &\text{For } k = 2: \quad 253 \text{ pairs}, \\
   &\text{For } k = 3: \quad 253 \text{ pairs}, \\
   &\text{For } k = 4: \quad 252 \text{ pairs}, \\
   &\text{For } k = 5: \quad 252 \text{ pairs}, \\
   &\text{For } k = 6: \quad 251 \text{ pairs}, \\
   &\text{For } k = 7: \quad 251 \text{ pairs}.
   \end{aligned}
   \]
   Summing these pairs gives:
   \[
   254 + 254 + 253 + 253 + 252 + 252 + 251 + 251 = 2020.
   \]

6. **Conclusion:**
   The total number of ways to select the coins such that their sum is 2020 is exactly 2020.

The final answer is \(\boxed{2020}\).