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In combinatorial probability, understanding the combination formula is key to solving problems where order does not matter. The combination formula helps determine how many ways you can choose items from a larger set. It's denoted as \( \binom{n}{r} \), where \( n \) is the total number of items, and \( r \) is the number of items to be chosen. The formula for combinations is \[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \] This formula calculates how many possible sets you can form. For example, in part (a) of our exercise, the employee needs to choose 6 forms out of 10. Applying the combination formula: \[ \binom{10}{6} = \frac{10!}{6!(10-6)!} = 210 \] This represents all possible sets of 6 forms.

• Factorials (\(!\)) are repeated product sequences of integers less than or equal to \( n \).
• The order of selection is irrelevant, making it a combination, not a permutation.

Using combinations allows you to focus purely on the selection process without considering order.

Probability calculations in the given exercise rely on understanding favorable outcomes versus total possible outcomes. In probability, each experiment consists of a sample space with possible outcomes. Probability is calculated using the formula: \[ P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \] For part (a), the issue is determining the probability that only one type of form is left.

• Total combinations possible: \( \binom{10}{6} = 210 \).
• Favorable scenarios: All 6 withdrawal petitions selected (\( \binom{6}{6} \)) or all 4 substitutions selected (\( \binom{4}{4} \)).
  • Each gives only 1 way, resulting in 2 favorable outcomes in total.
• Calculation: \( P = \frac{2}{210} = \frac{1}{105} \).

In part (b), we focus on selecting forms so each successive is different. We found valid sequences: WSWS and SWSW. Each has 360 favorable ways, totaling 720 favorable outcomes versus 5040 total sequences possible. Thus, \[ P = \frac{720}{5040} = \frac{1}{7} \] Effectively, probability calculations turn complex scenarios into manageable numbers.

Approaching combinatorial problems requires a systematic methodology. Here is a step-by-step approach:
1. **Understand the Problem:** Break down what is being asked. Identify known values and variables.

2. **Determine Total Combinations:** Use the combination formula to identify the total number of outcomes. For the exercise, calculating \( \binom{10}{6} \) gives the total selections.

3. **Identify Favorable Cases:** Look for specific outcomes that meet the problem's criteria. For instance, isolating all one form type or alternating sequences.

4. **Calculate Probability:** Divide the number of favorable outcomes by the total possible outcomes, applying the fundamental probability formula.

5. **Verify Results:** Cross-check calculations for accuracy. Ensure the logic and math consistently reflect the problem setup.

This structured process fosters clear thinking and accuracy in dealing with probability and combinations.