To solve the problem, we need to evaluate both statements regarding the selection of four numbers from the set \( S = \{1, 2, 3, \ldots, 20\} \) and their arrangement to form an arithmetic progression (AP).
### Step 1: Total Ways to Choose 4 Numbers
The total number of ways to choose 4 numbers from a set of 20 is given by the combination formula:
\[
\text{Total ways} = \binom{20}{4}
\]
Calculating this:
\[
\binom{20}{4} = \frac{20!}{4!(20-4)!} = \frac{20 \times 19 \times 18 \times 17}{4 \times 3 \times 2 \times 1} = 4845
\]
### Step 2: Finding Favorable Outcomes for AP
Next, we need to find the number of ways to select 4 numbers that can form an AP.
1. **Common Difference \( d = 1 \)**:
- The sequences can be: \( (1, 2, 3, 4), (2, 3, 4, 5), \ldots, (17, 18, 19, 20) \)
- The first term can be \( 1 \) to \( 17 \) (17 sequences).
2. **Common Difference \( d = 2 \)**:
- The sequences can be: \( (1, 3, 5, 7), (2, 4, 6, 8), \ldots, (14, 16, 18, 20) \)
- The first term can be \( 1 \) to \( 14 \) (14 sequences).
3. **Common Difference \( d = 3 \)**:
- The sequences can be: \( (1, 4, 7, 10), (2, 5, 8, 11), \ldots, (11, 14, 17, 20) \)
- The first term can be \( 1 \) to \( 11 \) (11 sequences).
4. **Common Difference \( d = 4 \)**:
- The sequences can be: \( (1, 5, 9, 13), (2, 6, 10, 14), \ldots, (8, 12, 16, 20) \)
- The first term can be \( 1 \) to \( 8 \) (8 sequences).
5. **Common Difference \( d = 5 \)**:
- The sequences can be: \( (1, 6, 11, 16), (2, 7, 12, 17), (3, 8, 13, 18), (4, 9, 14, 19), (5, 10, 15, 20) \)
- The first term can be \( 1 \) to \( 5 \) (5 sequences).
Adding these favorable outcomes:
\[
\text{Total favorable outcomes} = 17 + 14 + 11 + 8 + 5 = 55
\]
### Step 3: Probability Calculation
Now, we can calculate the probability that the chosen numbers will form an AP:
\[
\text{Probability} = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{55}{4845}
\]
Simplifying this fraction:
\[
\frac{55}{4845} = \frac{1}{85}
\]
### Conclusion for Statement 1
Thus, **Statement 1** is correct: The probability that the chosen numbers when arranged in some order will form an AP is \( \frac{1}{85} \).
### Step 4: Evaluating Statement 2
**Statement 2** claims that the set of all possible values of the common difference \( d \) is \( \{1, 2, 3, 4, 5\} \).
From our analysis, we found that the common differences can be \( 1, 2, 3, 4, 5 \), but we also need to check if there are any other possible values. The maximum common difference that can be used while still allowing for four terms to be selected from the set \( S \) is indeed \( 5 \) (as shown in the calculations).
Thus, **Statement 2** is also correct.
### Final Answer
- **Statement 1**: Correct
- **Statement 2**: Correct
