Problem 89 - Precalculus with Limits

Question

What is the relationship between \( _nC_r \) and \( _nC_{n - r} \)?

Step-by-Step Solution

Answer

The relationship between \( _nC_r \) and \( _nC_{n - r} \) is that they are equal. The number of ways to select r items from a set of n items is the same as the number of ways to select \(n - r\) items from the same set.

Step 1: Understand Combinations Formula

The formula for combinations is given by \( _nC_r = \frac{n!}{r!(n - r)!}\) where \(n!\) is the factorial of the total number of items, \(r!\) is the factorial of the number of chosen items, and \((n - r)!\) is the factorial of the difference between the two.

Step 2: Derive the reciprocal relation

Replace r in the formula by \(n - r\). This gives us \( _nC_{n-r} = \frac{n!}{(n-r)!(n - (n - r))!}\). Simplifying the equation gives us \( _nC_{n-r} = \frac{n!}{(n - r)!r!}\).

Step 3: Compare the equations

When we compare the equations derived in Step 1 and Step 2, it's apparent that \( _nC_r \) is equal to \( _nC_{n-r} \). The r objects selected from n objects are equivalent to the \(n - r\) objects not selected.

Key Concepts

Factorials

Factorials are a fundamental concept in combinatorics and play a key role in calculating permutations and combinations. Mathematically, a factorial is denoted by an exclamation mark, such as \( n! \). The factorial of a number \( n \) is the product of all positive integers less than or equal to \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). Factorials grow rapidly with larger numbers.
Understanding factorials is crucial in the context of combinations, as they help account for the total number of ways to arrange or choose items. For example:

• \( 0! \) is defined to be 1, as there is exactly one way to arrange nothing.
• Factorials are used in the denominator of the combinations formula to remove permutations within selected subsets.

Comprehending how factorials function is essential to mastering more complex combinatorial concepts.

Binomial Coefficients

Binomial coefficients are significant in the study of combinatorics as they count the number of ways to choose a subset of \( r \) elements from a set of \( n \) elements without considering the order. The notation \( _nC_r \) or \( \binom{n}{r} \) is often used to represent binomial coefficients. The formula for calculating these coefficients is given by:
\[_nC_r = \frac{n!}{r!(n - r)!}\]
This formula illustrates how factorials come into play to determine the number of combinations. The binomial coefficient is symmetric, meaning it satisfies the property:

• \( \binom{n}{r} = \binom{n}{n-r} \)

This symmetry indicates that choosing \( r \) elements is equivalent to leaving out \( n-r \) elements from the set, hence the relation \( _nC_r = _nC_{n-r} \) in the original problem.

Combinations

Combinations are a way to select items from a larger set such that the order of selection does not matter. This is different from permutations, where order does matter. When calculating combinations, the formula used is:
\[ _nC_r = \frac{n!}{r!(n-r)!} \]
where \( n \) is the total number of items and \( r \) is the number of items to choose. Using the combination formula, we ensure that the order in which items are selected is not counted multiple times. This is achieved through the factorial in the formula.
In practical terms, combinations are useful in situations such as:

• Selecting a subset of a group without ranking (e.g., choosing team members from a pool of candidates).
• Creating lists where the sequence of selection is irrelevant (e.g., choosing toppings for a pizza).

Understanding combinations allows us to solve problems where the focus is on choosing items rather than arranging them.