Problem 82 - A Graphical Approach to Precalculus with Limits

Question

Prove each statement for positive integers \(n\) and \(r\), with \(r \leq n\). (Hint: Use the definitions of permutations and combinations.) $$C(n, n-r)=C(n, r)$$

Step-by-Step Solution

Answer

The identity \(C(n, n-r) = C(n, r)\) is true because the expressions for both combinations are identical.

Step 1: Recall the definition of combinations

Combinations, denoted as \(C(n, r)\) or sometimes as \(\binom{n}{r}\), represent the number of ways to choose \(r\) elements from \(n\) elements without regard to order. The formula for combinations is given by:\[C(n, r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}\]

Step 2: Apply the given equation

We need to prove that \(C(n, n-r) = C(n, r)\). By the definition of combinations:\[C(n, n-r) = \frac{n!}{(n-r)!(r)!}\]Comparing this with \(C(n, r) = \frac{n!}{r!(n-r)!}\), we see both expressions are identical.

Step 3: Compare the two expressions

From Step 2, we notice:\[C(n, n-r) = \frac{n!}{(n-r)!(r)!} = \frac{n!}{r!(n-r)!} = C(n, r)\]Since both expressions are equal, the identity holds true. This demonstrates that the number of ways to choose \(n-r\) elements from \(n\) is the same as the number of ways to choose \(r\) elements from \(n\).

Key Concepts

Combinations

Combinations are a foundational concept in combinatorics and refer to selecting items from a greater set, where the order of selection does not matter. When dealing with combinations, we often use the notation \( C(n, r) \), also expressed as the binomial coefficient \( \binom{n}{r} \). This notation represents the number of ways to choose \( r \) elements from a set of \( n \) elements without paying attention to the sequence of selection.

To calculate the number of combinations, we use the formula:

• \( C(n, r) = \frac{n!}{r!(n-r)!} \)

In this formula, \( n! \) (n factorial) means multiplying all positive integers up to \( n \), \( r! \) means multiplying all positive integers up to \( r \), and \( (n-r)! \) covers the multiplication for the difference.

Understanding combinations is key to solving problems where the sequence of selection does not affect the outcome, like selecting a committee or choosing lottery numbers.

Permutations

Permutations involve arranging items from a set where the order does matter. This is different from combinations, where order is not considered. Permutations are useful when the arrangement of chosen items impacts the overall outcome.

The formula for permutations of \( r \) objects chosen from a set of \( n \) is:

• \( P(n, r) = \frac{n!}{(n-r)!} \)

This formula calculates the number of possible arrangements by taking the factorial of \( n \) and dividing it by the factorial of \( (n-r) \).

Permutations are used in scenarios like ranking contestants, forming a queue, or setting passwords where the position and order significantly matter. Knowing the difference between permutations and combinations can aid in tackling various problems efficiently.

Binomial Coefficient

The binomial coefficient is a central concept in combinatorics that is commonly seen in the "binomial theorem." It provides insight into how to expand expressions of the powers of sums, like \((a+b)^n\). The binomial coefficient is represented as \( \binom{n}{r} \) and calculated using the formula for combinations:

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

The binomial coefficient tells us how many ways \( r \) items can be selected from \( n \) items, which is why it aligns with the concept of combinations.

In the context of the binomial theorem, the binomial coefficients become the coefficients of each term in the expanded form of \((a + b)^n\). This application shows how powerful the binomial coefficient can be when exploring algebraic expressions and their expansions.

Utilizing the binomial coefficient effectively can simplify complex problems involving selections, distributions, and expansions in mathematics.