Problem 50 - Precalculus: Mathematics for Calculus

Question

Show that the given statement is true. $$\left(\begin{array}{l}n \\ 0\end{array}\right)=1\( and \)\left(\begin{array}{l}n \\ n\end{array}\right)=1$$

Step-by-Step Solution

Answer

Both \( \binom{n}{0} \) and \( \binom{n}{n} \) are equal to 1.

Step 1: Understand Binomial Coefficients

Binomial coefficients are coefficients that appear in the expansion of a binomial raised to a power. They are denoted as \( \binom{n}{k} \) and represent the number of ways to choose \( k \) elements from a set of \( n \) elements.

Step 2: Evaluate \( \binom{n}{0} \)

The formula for a binomial coefficient is \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \). When evaluating \( \binom{n}{0} \), we substitute \( k = 0 \) into the formula: \( \binom{n}{0} = \frac{n!}{0!(n-0)!} = \frac{n!}{n!} = 1 \). Thus, \( \binom{n}{0} = 1 \).

Step 3: Evaluate \( \binom{n}{n} \)

Similarly, for \( \binom{n}{n} \), substitute \( k = n \) into the formula: \( \binom{n}{n} = \frac{n!}{n!(n-n)!} = \frac{n!}{n! \cdot 0!} \). Since \( 0! = 1 \), this simplifies to \( \frac{n!}{n!} = 1 \). Hence, \( \binom{n}{n} = 1 \).

Step 4: Conclusion

Based on the calculations, both binomial coefficients \( \binom{n}{0} \) and \( \binom{n}{n} \) are equal to 1, which confirms the truth of the statement given in the exercise.

Key Concepts

Factorial function

The factorial function is a mathematical operation that multiplies a series of descending positive integers. It is denoted by an exclamation mark (!). For example, the factorial of a number \( n \), written as \( n! \), is the product of all positive integers from \( 1 \) to \( n \).
For instance:

• \( 5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120 \)
• \( 3! = 3 \cdot 2 \cdot 1 = 6 \)
• \( 0! = 1 \) (by definition)

This function is essential in many areas of mathematics, including combinatorics, as it helps in calculating permutations and combinations. Whenever you see a formula involving factorials, it's likely dealing with arrangements or selections of items.
Understanding the factorial function is crucial when you want to find how many different ways you can arrange or select objects from a group. Remember, \( 0! = 1 \) by definition is a key piece to resolve scenarios where no selections are made, like choosing no elements from a set.

Combinatorial mathematics

Combinatorial mathematics is a branch of mathematics concerning the counting, arrangement, and combination of objects. It seeks to answer questions like "How many ways can we arrange these objects?" or "How many ways can we choose a subset of these objects?"
One of the core concepts in combinatorics is the combination, which is a selection of items from a larger pool where the order does not matter. For example, selecting 2 fruits from a basket containing 3 distinct fruits.
Combinations are calculated using the binomial coefficient formula, which incorporates the factorial function:

• The formula is \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \), where \( n \) is the total number of items, and \( k \) is the number of items to choose.

This formula is often used in probability and statistics. Understanding combinatorial mathematics helps solve problems related to permutations and combinations efficiently.

Binomial theorem

The binomial theorem is a foundational principle in algebra that provides a way to expand expressions that are raised to a power. It permits expressions of the form \( (a + b)^n \) to be expanded into a sum involving terms of the form \( \binom{n}{k} a^{n-k} b^k \).
Key points about the binomial theorem include:

• It uses binomial coefficients \( \binom{n}{k} \) to determine the coefficients of the expanded terms.
• The theorem is represented as \( (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \).

Each term in the expansion corresponds to a particular number of occurrences of \( a \) and \( b \). The sum of the exponents in each term equals \( n \).
The binomial theorem not only simplifies calculations for high powers but also finds applications in probability, statistics, and many branches of mathematics. Understanding how to use the binomial theorem effectively allows for efficient computation of polynomial expansions and solutions to various mathematical problems.