Problem 48 - College Algebra

Question

A conductor needs 5 cellists and 5 violinists to play at a diplomatic event. To do this, he ranks the orchestra's 10 cellists and 16 violinists in order of musical proficie cy. What is the ratio of the total cellist rankings possible to the total violinist rankings possible?

Step-by-Step Solution

Answer

The ratio is approximately 1:17.

Step 1: Determine the Total Cellist Rankings

We have 10 cellists and need to choose 5 of them to play. The number of ways to do this is given by the combination formula: \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \). Here, \( n = 10 \) and \( r = 5 \). Thus, the total cellist rankings possible are \( \binom{10}{5} = \frac{10!}{5!5!} = 252 \).

Step 2: Determine the Total Violinist Rankings

For the violinists, we have 16 violinists, and we need to choose 5 of them. Therefore, the number of ways to choose them using the combination formula is \( \binom{16}{5} = \frac{16!}{5!11!} = 4368 \).

Step 3: Calculate the Ratio of Cellist Rankings to Violinist Rankings

Now, we need to find the ratio of the cellist rankings to the violinist rankings, which is given by the ratio \( \frac{\text{Total Cellist Rankings}}{\text{Total Violinist Rankings}} = \frac{252}{4368} \). Simplifying this ratio gives \( \frac{1}{17.333} \) or approximately \( \frac{1}{17} \).

Step 4: Simplify the Ratio

The exact calculation of the ratio \( \frac{252}{4368} \) simplifies further to the smallest integer ratio by dividing both terms by 252, which gives \( \frac{1}{17.333} \), which can approximately be considered as \( \frac{1}{17} \) since \( 17.333 \approx 17 \).

Key Concepts

Combination Formula

The combination formula is a vital tool in combinatorics that allows us to determine how many ways we can choose a subset from a larger set. For any set size \( n \) and a subset size \( r \), the number of ways to choose \( r \) elements is given by the formula:

• \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)

This formula helps us calculate combinations without regard to order. In the orchestra selection problem, we use this formula to determine how many ways the conductor can select 5 cellists from a group of 10 and 5 violinists from 16. Understanding this is essential when working on problems involving selections without replacement or order.

Factorial

Factorials, denoted by an exclamation point (!), are a fundamental concept in mathematics and combinatorics. A factorial of a number \( n \) is the product of all positive integers equal to or less than \( n \). Thus, the factorial of a number \( n \) is expressed as:

• \( n! = n \times (n-1) \times (n-2) \times \ldots \times 1 \)

This function is crucial in calculating combinations since it forms the backbone of the combination formula. For instance, in the orchestra selection problem, the factorials \( 10! \) and \( 5! \) help determine the total number of possible selections and rankings.

Ratios

A ratio is a way to compare two quantities by using division. It expresses how much of one thing there is compared to another. In the context of the orchestra selection problem, we aim to understand the comparative number of ways to select cellists versus violinists.

• The ratio formula is \( \frac{a}{b} \), where \( a \) and \( b \) are any two nonzero numbers.
• To simplify a ratio, divide both the numerator and the denominator by their greatest common divisor.

In this exercise, the cellists' combination result (252) is compared to the violinists' (4368), yielding a simplified ratio \( \frac{1}{17} \), indicating that there are 17 times more possible rankings for the violinists than for the cellists.

Orchestra Selection Problem

The orchestra selection problem is a real-world application of combinatorics where we need to form groups from larger sets. In this specific scenario, it involves selecting musicians for a performance based on their proficiency ranking. The key steps are:

• Identify the total number of musicians available in each category (cellists and violinists).
• Determine the number to be selected in each category.
• Use the combination formula to calculate the number of ways to select them.
• Compare the results using ratios to assess the difference in selection flexibility.

This type of problem showcases how combinatorics resolves practical decisions like forming an ensemble, ensuring the selections are optimized and ordered by rankings, making it quite relevant in musical and other selection-based scenarios.