Permutation vs. Combination: Key Rules and Example

Core idea: Permutations vs. Combinations

• In probability counting, first decide whether order matters before counting outcomes.

  • If order matters, you use permutations.
  • If order does not matter, you use combinations.

• Important rule: You cannot mix counting methods between the numerator and the denominator.

  • Either both are counted using permutations, or both are counted using combinations.
  • The transcript emphasizes: they either both have to be, like in our first two, the number one and a permutation or two permutations, two combinations.

• How to decide which to use:

  • Ask: Is order important for the event I’m counting? Do I need an ordered sequence or an unordered set?

• Example discussed in the transcript: Probability that they finish first and second.

  • Since we want them finishing first and second, the order is important.
  • Therefore, this is a permutation counting problem.

Key formulas

• Permutations (order matters):

  • General form: P(n,k) = rac{n!}{(n-k)!}
  • For the top 2 positions: P(n,2) = rac{n!}{(n-2)!} = n(n-1)

• Combinations (order does not matter):

  • General form: C(n,k) = inom{n}{k} = rac{n!}{k!(n-k)!}
  • For the top 2 positions: C(n,2) = rac{n!}{2!(n-2)!} = rac{n(n-1)}{2}

• Relationship between the two for k = 2:

  • $P(n,2) = 2 imes C(n,2)$
  • This shows that ordered counts double the unordered counts for two items selected from n.

Worked example: Top-two finishes among n contestants

• Total number of ordered top-two finishes (first and second) = $P(n,2) = n(n-1)$
• If you want a particular ordered pair (A finishes first, B finishes second):
  • Favorable outcomes = 1
  • Probability = rac{1}{P(n,2)} = rac{1}{n(n-1)}
• If you want either order for a given pair (A and B in the top two, in any order):
  • Favorable outcomes = 2
  • Probability = rac{2}{P(n,2)} = rac{2}{n(n-1)}
• If you counted the top-two as an unordered set (A and B in the top two, order ignored):
  • Number of outcomes = C(n,2) = rac{n(n-1)}{2}
  • Note: This is not appropriate for events where order (first vs second) matters.

Practical implications and intuition

• Always align the event description with the counting method:
  • If the event cares about position (who is first, who is second), use permutations for both numerator and denominator.
  • If the event cares only about which individuals are in the top group (not the order), use combinations for both.
• The total number of outcomes in the sample space depends on the counting method: permutations yield more outcomes than combinations by a factor of 2! for two items.
• The choice affects probabilities directly; mixing methods leads to incorrect probabilities.

Quick decision guide

• If you need an ordered sequence of k distinct items from n: use $P(n,k)$.
• If you only need a set of k items (order ignored): use $C(n,k)$.
• Ensure the denominator (total outcomes) uses the same counting method as the numerator (favorable outcomes).

Connections to foundational principles

• Sample space construction: choosing the right counting method defines the sample space.
• Uniform probability assumption: each counted outcome is assumed equally likely.
• Consistency between numerator and denominator is essential for valid probability calculations.