Problem 9 - A Graphical Approach to College Algebra

Question

Evaluate the following. In Exercises 17 and \(18,\) express the answer in terms of \(n .\) Do not use a calculator. $$\left(\begin{array}{c}13 \\\13\end{array}\right)$$

Step-by-Step Solution

Answer

The value of \( \left( \begin{array}{c} 13 \\ 13 \end{array} \right) \) is 1.

Step 1: Understanding the Problem

We need to evaluate the binomial coefficient \( \left( \begin{array}{c} 13 \ 13 \end{array} \right) \), which represents the number of ways to choose 13 items from 13 without regard to the order of selection.

Step 2: Applying the Binomial Coefficient Formula

The formula for a binomial coefficient is given by \( \left( \begin{array}{c} n \ r \end{array} \right) = \frac{n!}{r!(n-r)!} \). In this problem, \( n = 13 \) and \( r = 13 \).

Step 3: Substituting Values into the Formula

Substituting the values into the formula, we have \( \left( \begin{array}{c} 13 \ 13 \end{array} \right) = \frac{13!}{13!(13-13)!} = \frac{13!}{13! \cdot 0!} \).

Step 4: Simplifying the Expression

We know that \( 0! = 1 \). So the expression simplifies to \( \frac{13!}{13! \cdot 1} \). Divide \( 13! \) by itself to get \( 1 \).

Step 5: Conclusion

The binomial coefficient \( \left( \begin{array}{c} 13 \ 13 \end{array} \right) \) evaluates to \( 1 \), which indicates there is exactly one way to choose all 13 items from a set of 13.

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, arrangement, and combination of objects. It's like finding different ways to make groups from a larger set. A common problem in combinatorics is determining how many ways a certain number of items can be selected from a larger group without caring about the order in which they're selected. This is formally expressed using the concept of binomial coefficients.

In the problem at hand, we use combinatorial reasoning to determine the number of ways to choose 13 items from a set of 13. This scenario is a straightforward application because when the number to choose is the same as the set size, there's only one possible way to choose, as every item is selected. Thus, it highlights the simplicity and sometimes intuitive nature of combinatorial problems.

Factorial Notation

Factorials are a key part of combinatorics and are denoted by the exclamation mark (!). The factorial of a number, say n, written as \( n! \), is the product of all positive integers up to n. For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). Factorials grow very quickly as numbers get larger, and they are fundamental in calculating permutations and combinations.

In solving binomial coefficients, factorials help in organizing and calculating the large numbers that arise when dealing with combinations. For instance, the expression for the binomial coefficient \( \binom{n}{r} \) involves factorials of n, r, and \( n - r \). A very important identity to remember is that \( 0! \) is defined as 1, which is essential in simplifying expressions where all items are chosen from the set.

Probability and Statistics

Probability and statistics often utilize combinatorial principles to handle various problems, especially where calculations of odds or likelihoods are concerned. The binomial coefficient is particularly useful in probability because it helps with calculating the number of favorable outcomes when order does not matter.

In our example, finding that there is only "one way" to choose 13 from 13 tells us about certainty in probability — when you're choosing all items from the set, you're certain to select them in only one way. This basic understanding sets the foundation for more complex probability problems, such as those involving larger sets or introducing additional conditions. Using combinatorics and binomial coefficients, probability questions can be framed in terms of counting combinations to determine the likelihood of certain selections or arrangements.