Problem 61 - Algebra and Trigonometry

Question

A basketball team has 6 players who play guard ( 2 of 5 starting positions). How many different teams are possible, assuming that the remaining 3 positions are filled and it is not possible to distinguish a left guard from a right guard?

Step-by-Step Solution

Answer

There are 15 different ways to choose the 2 guards.

Step 1: Understanding the Problem

Identify what is being asked. The problem wants to find the number of different ways to select 2 guards out of 6 players, where the positions of left guard and right guard are not distinguishable.

Step 2: Use Combinations

Given we need to choose 2 players from 6 without regard to order, use the combination formula \({n \choose k} = \frac{n!}{k!(n-k)!}\). Here, n = 6 and k = 2.

Step 3: Calculate the Combination

Substitute n=6 and k=2 into the combination formula: \[ \binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6!}{2!\,4!} \]

Step 4: Simplify the Factorials

Simplify the expression by calculating the factorials: \[ \binom{6}{2} = \frac{6 \times 5 \times 4!}{2 \times 1 \times 4!} = \frac{30}{2} = 15 \]

Step 5: Interpret the Result

There are 15 different ways to choose 2 guards out of 6 players for the team, considering the positions are indistinguishable.

Key Concepts

combinations in probability

When dealing with problems like selecting team members, combinations are a key concept in probability. Combinations help us count the number of ways to choose items from a larger set, where the order doesn't matter. For example, if you have 6 players and need to choose 2 guards, you use combinations because it doesn't matter who is picked first, only who is picked in the end.
To calculate combinations, we use the combination formula: \({n \binom{k}} = \frac{n!}{k!(n-k)!}\). Here, \({n \binom{k}}\) denotes the number of combinations, \(!\) represents factorials, and \(n\) is the total number of items to choose from, while \(k\) is the number of items being chosen.

factorials

Factorials are essential in combinations and other areas of mathematics; they help us calculate the total number of ways to arrange a set of items. A factorial is the product of all positive integers up to a certain number. For instance, \(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\).
In our example, you're required to use factorials to determine the number of ways to choose 2 players out of 6. Remember, factorials grow very quickly, which is why they are useful for counting large sets efficiently.

binomial coefficient

The binomial coefficient is another term for combinations. It's represented as \(\binom{6}{2}\) in our case, which means '6 choose 2'. It's used to find the number of ways to pick \(k\) items from \(n\) items without regard to order.
Calculating \(\binom{6}{2}\) involves substituting into the combination formula: \(\binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6!}{2!4!}\). Simplifying this, we get \(\frac{6 \times 5 \times 4!}{2 \times 1 \times 4!} = \frac{30}{2} = 15\). Thus, there are 15 ways to select 2 guards from 6 players.

team selection

Team selection often involves combinatorics to figure out the number of possible team arrangements. In our example, we needed to choose 2 guards out of 6 players.
The process started with understanding that the positions of guards are not distinguishable (left or right guard doesn't matter). This simplifies the problem to a combination problem. Using the combination formula, we found \( \binom{6}{2} \) which equals 15. This means there are 15 different ways to choose the 2 guards. These kinds of methods can be applied to other team selection problems where the order of selection is not important.