Problem 51 - Precalculus: Mathematics for Calculus

Question

Show that the given statement is true. $$\left(\begin{array}{l}n \\ 1\end{array}\right)=\left(\begin{array}{c}n \\\ n-1\end{array}\right)=n$$

Step-by-Step Solution

Answer

The statement is true: both binomial coefficients equal to \( n \).

Step 1: Understand the Statement

The problem asks us to show that two binomial coefficients are equal, both of which are supposed to equal to \( n \). Specifically, we need to prove \( \binom{n}{1} = n \) and \( \binom{n}{n-1} = n \).

Step 2: Review the Binomial Coefficient Formula

The binomial coefficient \( \binom{n}{k} \) is given by the formula: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] where \(!\) denotes factorial, which is the product of all positive integers up to that number.

Step 3: Calculate \( \binom{n}{1} \)

Using the formula for the binomial coefficient, substitute \( n \) for \( n \) and \( 1 \) for \( k \):\[ \binom{n}{1} = \frac{n!}{1!(n-1)!} = \frac{n \cdot (n-1)!}{1 \cdot (n-1)!} = n \] The \((n-1)!\) in the numerator and denominator cancels out, leaving us with \( n \).

Step 4: Calculate \( \binom{n}{n-1} \)

Similarly, substitute \( n \) for \( n \) and \( n-1 \) for \( k \):\[ \binom{n}{n-1} = \frac{n!}{(n-1)!1!} = \frac{n \cdot (n-1)!}{(n-1)! \cdot 1} = n \] Again, the \((n-1)!\) terms cancel out, leaving just \( n \).

Step 5: Conclude the Proof

Having calculated both binomial coefficients, we found \( \binom{n}{1} = n \) and \( \binom{n}{n-1} = n \). Thus, we have proven the statement \( \binom{n}{1} = \binom{n}{n-1} = n \) to be true.

Key Concepts

Binomial Coefficient

When we talk about binomial coefficients, we refer to the simple and powerful tools used in combinatorics to calculate the number of ways to choose elements from a larger set. This is crucial in determining combinations, without the concern of order.

• The notation for a binomial coefficient is \( \binom{n}{k} \). This means we are choosing \( k \) elements from a total of \( n \) elements.
• For example, if you want to pick 2 fruits from a basket of 5 different fruits, the binomial coefficient formula helps find how many different pairs you can choose.

To understand why \( \binom{n}{1} = \binom{n}{n-1} = n \), let's dive into the expressions.

• \( \binom{n}{1} \) represents the number of ways to select 1 item from \( n \) total items, which is clearly \( n \).
• Similarly, \( \binom{n}{n-1} \) is the number of ways to exclude just one item from \( n \), leaving \( n-1 \) items selected, and thus also equals \( n \).

This shows, through simple counting principles, these two coefficients are indeed equal and equal to \( n \).

Factorial Notation

Factorials are symbols in mathematics represented by the exclamation mark \((!)\) that denote the product of all positive integers up to a given number. This notation simplifies the calculation of combinatory processes, especially in binomial coefficients.

• For a number \( n \), the factorial is \( n! = n \times (n-1) \times (n-2) \times ... \times 1 \).
• By definition, \( 0! \) is \( 1 \), which often helps in simplifying calculations involving combinatorial mathematics.

Factorials help to reduce complex expressions to easy-to-calculate formulas. For example:

• Calculating \( \binom{n}{1} = \frac{n!}{1!(n-1)!} \), we see that \( 1! \) and \( (n-1)! \) simplify down, leaving \( n \).
• Similarly, \( \binom{n}{n-1} = \frac{n!}{(n-1)!1!} \) simplifies in the same way to \( n \).

These simplifications through factorials provide clarity and simplicity to the handling of binomial calculations.

Proof Techniques

Proof techniques are fundamental in mathematics to validate the truth of statements. By meticulously breaking down problems with clear logical steps, you ensure the integrity of your solution.

• A common technique is direct proof, where you follow a straightforward path from assumptions to conclusion, just as in our step-by-step solution.
• We began by identifying what needed to be shown – specifically, that two binomial coefficients equal \( n \).

In our exercise, we used a specific approach:

• First, apply the known formula for binomial coefficients \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \) to calculate the equalities.
• Second, simplify these expressions using the properties of factorials, such as \((n-1)!\) canceling out in the numerator and denominator.
• Conclude with the observation that both binomial expressions indeed result in \( n \), satisfying the initial statement to be proven.

Using proof techniques thoughtfully helps bridge the gap between understanding a problem and demonstrating its solution, aiding clarity and confidence in mathematical reasoning.