Problem 21 - Calculus for Biology and Medicine

Question

A committee of 3 people must be formed from a group of 10. How many committees can there be if no specific tasks are assigned to the members?

Step-by-Step Solution

Answer

There are 120 different committees possible.

Step 1: Define the Problem

We need to determine how many different committees of 3 people can be formed from a group of 10 people. Importantly, the roles within the committee are not distinct, meaning order does not matter.

Step 2: Identify the Formula for Combinations

To solve this problem, we use the combinations formula, which calculates how many ways we can choose a subset of items from a larger set without regard to the order. The formula is given by: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]where \( n \) is the total number of items to choose from, and \( k \) is the number of items to choose.

Step 3: Substitute Values into the Formula

Here, \( n = 10 \) and \( k = 3 \). Substituting these values into the combinations formula gives us:\[ \binom{10}{3} = \frac{10!}{3!(10-3)!} = \frac{10!}{3!\times7!} \]

Step 4: Calculate the Factorials

Calculate the factorials:- \( 10! = 10 \times 9 \times 8 \times 7! \)- \( 3! = 3 \times 2 \times 1 = 6 \)- \( 7! = 7! \)Substituting back into the equation gives:\[ \binom{10}{3} = \frac{10 \times 9 \times 8 \times 7!}{6 \times 7!} \]

Step 5: Simplify the Expression

Cancel out \( 7! \) from the numerator and denominator:\[ \frac{10 \times 9 \times 8}{6} \]Now, simplify this expression step-by-step. First, divide 9 by 3 which gives 3:\[ \frac{10 \times 3 \times 8}{2} \]Then divide 8 by 2 which gives 4:\[ 10 \times 3 \times 4 \]

Step 6: Compute the Final Product

Calculate the remaining multiplication:\[ 10 \times 3 = 30 \]\[ 30 \times 4 = 120 \]

Step 7: Conclusion

Thus, there are 120 different ways to form a committee of 3 people from a group of 10 without regard to order.

Key Concepts

Understanding Factorials

Factorials are a fundamental concept in combinatorics. They represent the product of a number and all the integers below it, down to one. Factorials are denoted by an exclamation mark (!). For example:

• 3! = 3 × 2 × 1 = 6
• 4! = 4 × 3 × 2 × 1 = 24
• 5! = 5 × 4 × 3 × 2 × 1 = 120

Factorials are used in many mathematical formulas, particularly in combinatorics, because they simplify the calculations involving large sets of numbers. They express how many ways you can arrange a set of items.
In this context, they help determine how many specific arrangements are possible by accounting for all sequential variations. Understanding how to compute and use factorials is essential for navigating through more complex math problems involving permutations and combinations.

Combinations Formula Explained

When determining how many ways we can choose a group of items where order doesn't matter, the combinations formula is the best tool for the job. The combinations formula is expressed as:\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]Here:

• \(n\) stands for the total number of items available.
• \(k\) is the number of items we want to choose.

The formula derives from the need to find the total possible arrangements of a subset, where order is not a concern. Imagine you have 10 people and you want to choose 3 to form a committee. Here, \(n = 10\) and \(k = 3\), so you substitute these values into the formula to calculate:\[\binom{10}{3} = \frac{10!}{3!(10-3)!}\]Which breaks down further to \(\frac{10!}{3!7!}\). This calculation process helps simplify situations by determining the exact number of ways to select the required items, without being concerned about different sequences.

Subset Selection Simplified

Subset selection is choosing a specific number of elements from a larger set. This task is common in scenarios where only a few items from a bigger pool are needed, and their order doesn't matter. The concept of subset selection is tightly linked to combinations.
To truly grasp subset selection, focus on these main points:

• The total number of ways to choose subsets is calculated using the combinations formula.
• In the given exercise, selecting 3 people out of 10 means you're choosing an unordered subset.
• Analyzing subsets helps in recognizing different potential groupings or selections within a larger group.

This simple approach gives you an accurate count of possible groupings, informed by a solid understanding of combinations. This method is versatile across a range of activities, from forming committees to organizing teams or events, providing critical insight wherever selection is key.