Problem 76 - Algebra and Trigonometry

Question

Solve the problem using the appropriate counting principle(s). Choosing a Delegation Three delegates are to be chosen from a group of four lawyers, a priest, and three professors. In how many ways can the delegation be chosen if it must include at least one professor?

Step-by-Step Solution

Answer

There are 46 ways to choose a delegation with at least one professor.

Step 1: Determine Total Number of Delegations

First, calculate the total number of ways to choose 3 delegates from the entire group of 8 people (4 lawyers, 1 priest, and 3 professors). Use the combination formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \). Here \( n = 8 \) and \( r = 3 \). \[ \binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56 \]There are 56 total ways to choose 3 delegates.

Step 2: Calculate Delegations with No Professors

Now calculate the number of ways to choose 3 delegates such that none are professors (i.e., from the 4 lawyers and 1 priest only). This means choosing 3 from the group of 5. \[ \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10 \]There are 10 ways to choose 3 delegates with no professors.

Step 3: Subtract Non-Professor Delegations From Total

To find the delegations that include at least one professor, subtract the number of non-professor delegations from the total number of delegations:\[ 56 - 10 = 46 \]Therefore, there are 46 ways to choose a delegation that includes at least one professor.

Key Concepts

Combination Formula

When solving problems that involve selecting a group from a larger set, the combination formula is a key tool. The formula is written as \( \binom{n}{r} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose. The formula calculates the number of unique combinations possible, accounting for the fact that order does not matter in combinations. It is given by

• \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)
• Where \( n! \) (n factorial) is the product of all positive integers up to \( n \)

Factorials simplify the understanding of permutations and combinations, but it is important to remember this formula when order isn't important. By applying the combination formula, one can swiftly compute the number of ways to choose subsets within a set, with direct application in countless combinatorial challenges.

Counting Principle

The counting principle is a fundamental strategy in combinatorics to determine the number of possible outcomes. It helps us break down complex problems into manageable parts by examining how choices accumulate. It can be explained as follows:

• **Multiplication Rule:** If a task consists of several stages, the total number of ways to carry out the task is the product of the number of ways to complete each stage.
• This principle forms the basis for more complex counting methods, like permutations and combinations.

In the context of selecting delegations, the counting principle allows you to calculate the total number of possible combinations in a systematic way. Its key utility lies in simplifying matters by ensuring all possibilities are covered without repetition.

Delegation Selection

In scenarios like delegation selection, often, we want conditions to be met. For example, ensuring that at least one professor is included in the delegation. To factor such conditions into our calculations, we may employ techniques such as the complementary counting method:

• **Complementary Counting:** Calculate all possible selections, and subtract those that don't meet the condition (e.g., no professors) from this total.
• By methodically excluding what doesn't meet the criteria, we are left only with the desired outcomes.

This method is practical within the framework of the combination formula. It elegantly handles scenarios by eliminating the unwanted cases rather than dealing directly with multiple restrictive conditions. This approach is not only logical but also efficient, making it a favorite among mathematicians and students alike.