Problem 56 - Algebra and Trigonometry

Question

A defensive football squad consists of 25 players. Of these, 10 are linemen, 10 are linebackers, and 5 are safeties. How many different teams of 5 linemen, 3 linebackers, and 3 safeties can be formed?

Step-by-Step Solution

Answer

302400 different teams.

Step 1: Calculate linemen combinations

To find the number of ways to choose 5 linemen out of 10, use the combination formula \[ C(n, k) = \frac{n!}{k!(n-k)!} \]Plugging in the values: \[ C(10, 5) = \frac{10!}{5!(10-5)!} = \frac{10!}{5!5!} \]Simplifying further: \[ C(10, 5) = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} = 252 \]

Step 2: Calculate linebacker combinations

Similarly, to find the number of ways to choose 3 linebackers out of 10, again use the combination formula: \[ C(10, 3) = \frac{10!}{3!(10-3)!} = \frac{10!}{3!7!} \]Simplifying further: \[ C(10, 3) = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 \]

Step 3: Calculate safety combinations

Now, to find the number of ways to choose 3 safeties out of 5, use the combination formula: \[ C(5, 3) = \frac{5!}{3!(5-3)!} = \frac{5!}{3!2!} \]Simplifying further: \[ C(5, 3) = \frac{5 \times 4}{2 \times 1} = 10 \]

Step 4: Multiply all combinations

Finally, to get the total number of different teams, multiply the number of combinations for linemen, linebackers, and safeties: \[ 252 \times 120 \times 10 = 302400 \]

Key Concepts

Binomial Coefficients

Binomial coefficients are a key concept in combinatorics that help us calculate combinations.
Combinations are ways to select items from a larger pool without considering the order.
The binomial coefficient, often written as \( C(n, k) \) or \( \binom{n}{k} \), is calculated using the combination formula:

\[ C(n, k) = \frac{n!}{k!(n-k)!} \]
In this formula, \( n \) represents the total number of items, and \( k \) is the number of items to choose.
The exclamation mark \( ! \) denotes a factorial, which we'll explain next.

Factorials

Factorials are mathematical functions that multiply a positive integer by all the positive integers less than it.
It's represented by an exclamation mark \( ! \).
For example:

\ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \
Factorials are essential in combinatorics, particularly in calculating permutations and combinations.
In our exercise, we made use of factorials to determine the binomial coefficients for selecting players in different positions.

Combinatorics

Combinatorics is a field of mathematics that focuses on counting, arrangement, and combination of objects.
It helps answer questions like how many different ways we can arrange or select items.
In team selection exercises like ours, combinatorics is used to figure out how many ways we can select players from groups of linemen, linebackers, and safeties.
We used the combination formula to find out the possible selections for each subgroup, and then multiplied these solutions to get the total number of combinations.

Team Selection

Team selection is a common problem in combinatorics where we need to determine the number of ways to form a team from a group of candidates.
In our exercise, we have different groups of players—linemen, linebackers, and safeties—and we used binomial coefficients to find how many ways we can select the required number of players from each group.

Step-by-step:

• First, calculate the number of ways to select 5 linemen out of 10.
• Second, calculate the combinations for choosing 3 linebackers out of 10.
• Third, determine the ways to pick 3 safeties out of 5.
• Finally, multiply all the combinations together to get the total number of possible teams.

These methods ensure we consider all possible team combinations efficiently.