Problem 77 - Algebra and Trigonometry

Question

An Identity Involving Combinations Kevin has ten different marbles, and he wants to give three of them to Luke and two to Mark. How many ways can he choose to do this? There are two ways of analyzing this problem: He could first pick three for Luke and then two for Mark, or he could first pick two for Mark and then three for Luke. Explain how these two viewpoints show that $$C(10,3) \cdot C(7,2)=C(10,2) \cdot C(8,3)$$ In general, explain why $$C(n, r) \cdot C(n-r, k)=C(n, k) \cdot C(n-k, r)$$

Step-by-Step Solution

Answer

Both methods count ways to distribute marbles, leading to the identity \( C(10,3)\cdot C(7,2)=C(10,2)\cdot C(8,3) \).

Step 1: Understanding Combinations

Combinations refer to selecting items from a group, where the order does not matter. The notation \( C(n, r) \) represents the number of ways to choose \( r \) items from a total of \( n \) items.

Step 2: Breaking Down the Problem

Kevin wants to give 3 marbles to Luke and 2 to Mark. We can choose 3 marbles from 10 for Luke, and then 2 marbles from the remaining 7 for Mark. This is represented by the expression \( C(10,3) \times C(7,2) \).

Step 3: Reversing the Selection Process

Alternatively, Kevin could choose 2 marbles from 10 for Mark first, and then choose 3 marbles from the remaining 8 for Luke. This is expressed as \( C(10,2) \times C(8,3) \).

Step 4: Equivalence of Counting Methods

These two expressions show different ways to distribute the marbles, but they must yield the same total number of ways, emphasizing that \( C(10,3) \times C(7,2) = C(10,2) \times C(8,3) \). Each expression calculates a sequence of independent choices to achieve the same outcome.

Step 5: Generalizing the Identity

Generally, \( C(n, r) \times C(n-r, k) \) is the number of ways to pick \( r \) items and then \( k \) from the remainder, and \( C(n, k) \times C(n-k, r) \) is for choosing \( k \) items first and then \( r \) from what's left. Both structures count the same distinct outcome over the total \( n \) items, confirming the identity.

Key Concepts

Combinations

Combinations are a way to count the selection of items, where the arrangement or order of the items does not matter. This concept is vital in many areas of mathematics and real-world scenarios. If you think about choosing marbles from a bag, whether red, blue, or green, the focus is on which marbles are chosen, not the sequence. The mathematical notation for combinations is represented as \( C(n, r) \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items selected.

• \( C(n, r) \) is also referred to as "n choose r."
• For instance, \( C(10, 3) \) means how many ways can you choose 3 items out of 10.

Base these calculations on the formula:\[ C(n, r) = \frac{n!}{r! (n-r)!} \] Here, \(!\) is a factorial, which means multiplying a series of descending whole numbers. Understanding these basics helps tackle problems like determining how many ways Kevin can distribute marbles to his friends.

Binomial Coefficients

In the realm of combinatorics, binomial coefficients appear prominently in the expansion of binomials, that is expressions like \((a + b)^n\). These coefficients are essentially the same as combinations.A binomial coefficient is expressed as \( C(n, r) \) or \( \binom{n}{r} \), and it tells you how many different ways you can pick \( r \) elements from a set of \( n \) elements. It forms a crucial part of the binomial theorem: \[(a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r\]

• It helps calculate terms in polynomial expansions.
• \( \binom{n}{r} \) is determined by the same formula for combinations.

In the exercise problem, these coefficients help efficiently determine how many ways Kevin can give marbles to different people, without manually listing them all. They also play a role in validating identities like \( C(n, r) \cdot C(n-r, k) = C(n, k) \cdot C(n-k, r) \).

Counting Principles

Counting principles are foundational ideas used to count how many different ways an event can occur. These principles often involve combinations, permutations, and factorials. When Kevin decides to divide marbles between friends, he applies these principles by choosing marbles in steps:

• Firstly, by choosing how many marbles Luke receives, calculated using combinations.
• Secondly, selecting from the remaining marbles for Mark.

This demonstrates the basis of the Fundamental Principle of Counting. Where you can first select items for Luke followed by Mark or vice versa, ultimately arriving at the same result. Both methods count paths leading to the same outcome, which is why the mathematical identity \( C(10, 3) \cdot C(7, 2) = C(10, 2) \cdot C(8, 3) \) holds true.By understanding and applying these concepts, anyone can determine outcomes efficiently and verify identities across various mathematical contexts.