Problem 37 - Calculus for Biology and Medicine

Question

In how many ways can you form a committee of three people from a group of seven if two of the people do not want to serve together?

Step-by-Step Solution

Answer

There are 30 ways to form the committee without the two people serving together.

Step 1: Calculate Total Combinations

First, calculate how many ways you can choose 3 people out of 7 without any restrictions. This is calculated using combinations where you select 3 out of 7: \[ \binom{n}{r} = \binom{7}{3} = \frac{7!}{3!(7-3)!} \]When you compute this, you get:\[ \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \]So, there are 35 ways to select 3 people from 7.

Step 2: Calculate Restriction Scenarios

Next, identify the restriction scenario: if two people do not want to serve together, consider the pair as 'restricted' and calculate how many committees include both of these restricted people. Selecting the third member for the committee with these two restricted people can be done from the remaining 5 people:\[ \binom{5}{1} = 5 \] This is because we are choosing 1 additional person from the remaining 5.

Step 3: Subtract Restriction Scenarios from Total

Subtract the number of 'restricted' committees (where the two do not want to be together) from the total number of committees. Previously, we found there are 35 total ways and 5 restricted ways (including both people who don't want to work together):\[ 35 - 5 = 30 \]

Step 4: Final Step: Confirm the Result

After subtracting the restricted cases, we find there are 30 valid ways to form a committee of three people from the group of seven where the two specific people are not serving together.

Key Concepts

Combinations

Combinations are a fundamental concept in combinatorics, a branch of mathematics that focuses on counting and arranging. Combinations specifically deal with selecting items from a larger group where the order does not matter. For instance, if you want to choose 3 people out of a group of 7 to form a committee, the order in which you select them does not change the outcome. The formula for combinations is given by:\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]where:

• \( n \) is the total number of items to choose from
• \( r \) is the number of items to select

In our example, we use \( \binom{7}{3} \) to calculate combinations, which results in 35 different ways to form a committee of 3 out of 7 people. Remember, combination problems are all about the selection, not the order.

Permutations

Permutations also belong to the field of combinatorics, but unlike combinations, they are concerned with the arrangement of items where the order does matter. A permutation would be used if we were interested in arranging the committee members in a specific order, like assigning roles like chairperson, secretary, and treasurer. The formula for permutations is:\[ P(n, r) = \frac{n!}{(n-r)!} \]where:

• \( n \) is the total number of items to arrange
• \( r \) is the number of items you want to arrange in order

While permutations are not directly used in the problem of forming a committee in our original exercise, understanding the difference between combinations and permutations is vital. When choosing a committee, focus on combinations unless the order of selection or assignment of specific roles is required.

Restricted Choices

Restricted choices occur in combinatorics when certain conditions limit how selections can be made. These restrictions are crucial to consider after calculating initial combinations or permutations to ensure your final count only includes acceptable outcomes.In the original exercise, the restriction was that two of the seven people do not want to serve together on the committee. This required recalculating the combinations to exclude cases where those two people were selected together. To find the restricted combinations, you first calculate how many ways the two restricted people can be included in the committee: \[ \binom{5}{1} = 5 \]This calculation shows there are 5 ways to include both restricted people and choose one more from the remaining 5 individuals. By subtracting these 5 restricted committees from the total unrestricted combinations, we accounted for the preferences of people who do not want to serve together, resulting in 30 valid committee arrangements.By considering restrictions in combinatorial problems, you ensure the solution adheres to any specific conditions, leading to a more accurate result.