Question

A three out of five series is a competition between two teams consisting of at most five games and ending as soon as one of the two competing teams wins three games. How many different three out of five series are possible? Two series are "different" if the sequence of winners and losers in one series is not the same as in the other series. Draw a tree to represent the possibilities.

Short answer

There are 20 different possible series in a three out of five series.

Step-by-step solution

Understanding the Problem

The exercise asks us to find how many different sequences of games are possible in a three out of five series where one team must win three games to conclude the series. Each series can be up to five games long until one team wins three games.

Analyzing Possible Outcomes

The series can end in three, four, or five games. Each game contributes to reaching a possible sequence of wins and losses for one of the teams. We'll represent each game with either a 'W' for a win by the first team or 'L' for a win by the second team.

Calculating Series Ending in Three Games

The series ends in three games if one team wins all three. Possible sequences are: WWW for the first team and LLL for the second team. Thus, there are 2 outcomes for a series ending in exactly three games.

Calculating Series Ending in Four Games

In a four-game series, the first team wins three games while the second team wins one, or vice versa. The sequences are: WWWL, WLWW, LWWW for the first team, and LLWL, LWLL, WLLL for the second team. There are 6 outcomes for a series ending in exactly four games.

Calculating Series Ending in Five Games

In a five-game series, each team must win two games before the last game determines the series. Sequences are: WWLWW, WLWWW, LWLWW, WWLLW, WLWLW, LWWLW for the first team, and LLWWL, LWLWL, WLWWL, WWLLL, LWWLL, WLLWL for the second team. There are 12 outcomes for a series ending in exactly five games.

Adding All Possible Outcomes

We calculate the total number of outcomes by adding up the successful sequences for each series length: 2 (three games) + 6 (four games) + 12 (five games) = 20 different sequences possible in the series.

Drawing the Tree

Create a tree diagram starting from the first game with two branches (W or L) at each game, expanding similarly for each subsequent game up to five games. Highlight sequences where a team achieves three wins to signify the end of a series and check each unique path. Due to space constraints, diagramming a tree explicitly as text isn't feasible here, but starting with a game and branching at each decision will visualize potential outcomes.

Key concepts

Tree Diagrams

Tree diagrams are a fantastic way to visually map out all possible outcomes of a multi-step process, much like the three out of five series in our exercise. Imagine it as the branches of a tree, starting from a single trunk (representing the start of the series) and spreading out with each decision point, which in this case are the games won by each team.

Each game results in either a win (W) or a loss (L) for Team A. In the diagram, each match can be viewed as a node that branches into two: a win or a loss. As you move further down the tree, the branches continue to split until you have explored every potential ending of the series. The tree diagram helps us visualize scenarios where either team clinches three wins, signaling the end of a series.

• Start from the base of the tree and create branches for two possible outcomes (W or L) for the first game.
• Continue branching out from each node until a team wins three games.
• This visual approach ensures a comprehensive count of possible win/loss sequences.

The beauty of a tree diagram is that it not only highlights each potential sequence, but it also emphasizes how quickly outcomes can multiply with each new game.

Sequence Analysis

Sequence analysis involves scrutinizing the order in which events occur, in this case, how games are won in a series. Different sequences of wins and losses can result in distinct outcomes of the series, despite having the same team ultimately winning.

In our three out of five series, identifying how sequences can end in three, four, or five games is key to understanding the number of potential series outcomes. With each passing game, new possible sequences emerge, and by analyzing these, we can determine the sequence in which a team wins three games to end the series.

• "WWW" or "LLL" are sequences where a team wins in the shortest possible time, just 3 games.
• "WLWW" or "LWLW" illustrate slightly longer journeys to achieve victory, which finish in four games.
• In the longest format, five games bring sequences like "WWLWW" or "LLWLL" that showcase each team grasping at victory till the very end of the series.

By carefully analyzing each sequence and understanding the series' progressive nature, you build insight into strategic decision-making, illustrating how each game has pivotal repercussions on the outcome.

Permutations

Permutations are a fundamental concept in combinatorics which allow us to calculate the number of ways in which a sequence can be arranged. In a three out of five series, permutations help determine how different sequences of wins and losses can lead to a team winning three games.
To grasp permutations in this context, consider the fact that each series length (three, four, or five games) represents a distinct arrangement of wins and losses.

• A series ending in three games has only two permutations: "WWW" or "LLL" because there's just one way for each team to win three games consecutively.
• For a four-game series, combinations like "LWWW" take into account a single loss amid victories, resulting in 3 unique permutations for each team.
• A five-game series expands the permutation challenge, providing 6 possibilities per team, as each must win two initial games with the fifth deciding the winner.

Calculating permutations gives a comprehensive insight into all possible game outcomes. This emphasizes how different arrangements within the same game structure can lead to distinctive and varying series finales.