Final answer:
To find the recursion formula for the number of ways to pair off 2n people for tennis matches, we define P(n) as the number of pairings for 2n people and use the relation P(n) = (2n-1) × P(n-1) with the base case P(1) = 1.
Step-by-step explanation:
The student has requested a recursion formula for the number of ways to pair off 2n people for tennis matches. To start, assume that we have 2n players and we want to count the number of ways to organize them into n pairs. We take one person out of the group, and this person can choose their partner in 2n-1 ways. Once a partner is chosen, we are left with 2n-2 people to pair off. Therefore, the number of ways to pair these remaining people is the same as the problem for 2(n-1) people.
Let's define the function P(n) as the number of ways to pair off 2n people. According to the explanation above, P(n) can be recursively defined as P(n) = (2n-1) × P(n-1). To start the recursion, we note that when n=1 (so we have 2 people), there is only 1 way to pair them off. Therefore, P(1) = 1, which serves as our base case.
For full understanding, it's essential for students to practice recursion problems and grasp the concept of factorial calculation as it's a common procedure in combinatorics.