Problem 81 - 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. In 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 $$ C(n, r) \cdot C(n-r, k)=a(n, k) \cdot C(n-k, r) $$

Step-by-Step Solution

Answer

Both selections yield the same number of ways: \( C(10,3) \cdot C(7,2) = C(10,2) \cdot C(8,3) \).

Step 1: Analyze the problem

The problem involves distributing 10 different marbles between two people, Kevin and Mark. Kevin receives 3 marbles and Mark receives 2 marbles. Initially, consider the first approach: selecting 3 marbles for Kevin and then 2 for Mark.

Step 2: Calculate the combinations for first approach

To determine how many ways Kevin can receive 3 marbles, calculate the number of combinations for choosing 3 marbles out of 10. This is given by \( C(10,3) \). Following this selection, Mark chooses 2 marbles from the 7 that remain, which is \( C(7,2) \). Therefore, the total number of ways for this approach is \( C(10,3) \cdot C(7,2) \).

Step 3: Consider the alternate analysis

Now, consider selecting 2 marbles for Mark first. There are \( C(10,2) \) ways to do this. Then Kevin chooses 3 marbles from the remaining 8, given by \( C(8,3) \). The total number of ways in this approach is \( C(10,2) \cdot C(8,3) \).

Step 4: Establish equality of the two approaches

Both approaches essentially rearrange the process of selection but should result in the same number of combinations. By calculating, verify that \( C(10,3) \cdot C(7,2) = C(10,2) \cdot C(8,3) \).

Step 5: Explain the combinatorial identity

In general, for a set with \( n \) elements, selecting \( r \) elements first and then \( k \) elements can be described in terms of combinations. Thus, \( C(n,r) \cdot C(n-r,k) \) equals the number of ways to choose \( k \) elements first and then \( r \) elements from the remaining. This can be written as \( C(n,k) \cdot C(n-k,r) \). This equality holds due to the commutative nature of multiplication and the properties of combinations.

Key Concepts

Combinations

Combinations are a fundamental concept in combinatorics involving the selection of items from a larger set. They are employed when the order of selection doesn't matter, focusing solely on the group that is formed. For instance, when picking marbles from a bag, selecting three particular marbles results in the same combination, regardless of the order in which they are picked.

Think of it as making a group or a team: the members are key, not the sequence of picking them. In mathematical terms, the number of combinations of selecting 'r' items from a set of 'n' items is given by the binomial coefficient, represented as \( C(n, r) \).

Examples of combinations in daily life include forming committees, selecting lottery numbers, or distributing prize items among winners.

Binomial Coefficients

Binomial coefficients, represented as \( C(n, r) \), are a central concept in understanding combinatorial mathematics. They tell us the number of ways to choose 'r' elements from a set of 'n' elements without caring about the order of selection.

The formula for a binomial coefficient is: \[ C(n, r) = \frac{n!}{r!(n-r)!} \]

Here, \( n! \) (n factorial) is the product of all positive integers up to \( n \), and it highlights how richly we can rearrange elements. The denominator accounts for the repeated arrangements of the same group that don't count as distinct combinations.

Binomial coefficients appear in the binomial theorem, which extends to powers of a binomial expression, further linking the concept to broader mathematical principles.

Permutations

Permutations vary from combinations because they are all about the arrangement order. If combinations are about choosing the group, permutations deal with the way those group elements are organized. Therefore, for permutations, the sequence is crucial.

Suppose you have a scenario where there are three available positions, and you need to fill them with individuals from a larger group. Each new position could change the outcome, thus affecting the count.

The formula for permutations of selecting 'r' out of 'n' items is expressed as: \[ P(n, r) = \frac{n!}{(n-r)!} \]

This formula calculates the possible scenarios by allowing every selected item to match with every unavailable position, acknowledging different orders as different outcomes.

Commutative Property

In mathematics, the commutative property is a foundational principle that states you can change the order of numbers in an operation without changing the result. This concept applies both to addition and multiplication.

For multiplication, this means: \( a \times b = b \times a \).

When applying this to combinations, it explains why changing the order of selecting groups still gives rise to the same number of selections. For instance, when Kevin and Mark pick marbles in a different sequence, the final count of combinations remains unchanged because the total procedure is multiplicative and follows the commutative law.

Mathematical Proofs

Mathematical proofs play a significant role in establishing the truth of statements and identities in mathematics, building our understanding with rigor and logic. They rely on axioms, previously established theorems, and logical reasoning to conclude that a proposition is universally correct.

In the context of the given problem, a proof demonstrates why choosing marbles first in one way or another leads to the same number of outcomes.

Proofs often involve direct calculation and logical derivation, showing step-by-step that one side of an equation equals the other. They reassure students and mathematicians that concepts like combinatorial identities hold true in every attempt or scenario, providing confidence in their application and teaching.