Question

Show that \(\left(\begin{array}{c}n \\ n-1\end{array}\right)=n\) and \(\left(\begin{array}{l}n \\ n\end{array}\right)=1\).

Short answer

\( \binom{n}{n-1} = n \) and \( \binom{n}{n} = 1 \).

Step-by-step solution

- Understanding the Binomial Coefficient

The binomial coefficient is defined as \ \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\). It represents the number of ways to choose k elements from an n-element set.

- Calculating \(\binom{n}{n-1}\)

Start with the binomial coefficient formula: \(\binom{n}{n-1} = \frac{n!}{(n-1)!1!}\). Notice that \(1! = 1\). So, we get: \(\binom{n}{n-1} = \frac{n!}{(n-1)!} = n\) because the \(n!\) and \( (n-1)!\) terms cancel out leaving only the top number \( n \).

- Calculating \(\binom{n}{n}\)

Using the binomial coefficient formula again: \(\binom{n}{n} = \frac{n!}{n!0!}\). Since \(0! = 1\), it simplifies to: \(\binom{n}{n} = \frac{n!}{n!} = 1\).

- Conclusion

We've shown that \(\binom{n}{n-1} = n\) and \(\binom{n}{n} = 1\) using the definition of binomial coefficients and factorial cancellation.

Key concepts

Factorial

In mathematics, a factorial is denoted by an exclamation mark (!) and represents the product of all positive integers up to a given number. For example, 5! means multiplying 5 by all the smaller integers down to 1:
5! = 5 × 4 × 3 × 2 × 1 = 120.
Factorials are crucial in many areas of math, especially in combinatorics and probability.
They heavily feature in the binomial coefficient formula:
\(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).
When we compute \(n \binom{n}{n-1}\), we use the factorials to simplify the expression to just \( n! \) over \((n-1)!\), leading to just \(n\). Understanding how factorials work will help you simplify expressions effectively.

Combinatorics

Combinatorics is a branch of mathematics focused on counting, arrangement, and combination of objects.
It dives deep into figuring out how many ways you can organize objects under certain constraints.
The binomial coefficient, \(\binom{n}{k}\), is an essential concept in combinatorics representing the number of ways to choose \( k \) items from \( n \) items without considering order.
Think about it like this: if you want to choose 3 fruits from a set of 5 (e.g., apple, orange, banana, grape, kiwi), \(\binom{5}{3}\) gives you the count of different combinations you can make.
This is calculated using: \(\binom{5}{3} = \frac{5!}{3!(5-3)!} = 10\).
This fundamental idea helps in solving various complex problems related to counting and probability.

Binomial Theorem

The Binomial Theorem gives us a powerful way to expand expressions of the form \((a + b)^n\).
It states that: \((a + b)^n = \binom{n}{0}a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + \ldots + \binom{n}{n}b^n\).
Here, each term in the expansion uses the binomial coefficients.
For example, \((x + y)^2 \) expands to: \ x^2 + 2xy + y^2 \ ), where coefficients \(1, 2, and 1\) come from \( \binom{2}{0}, \binom{2}{1}, \) and \ (\binom{2}{2} resp.).
This theorem helps in breaking down complex expressions into manageable terms by using combinatorial principles.

Mathematical Proof

A mathematical proof is a logical argument that explains why a statement is true.
Proofs use previously established facts, like theorems and axioms, to show the truth of the statement beyond any doubt.
For example, proving \(\binom{n}{n-1} = n\) involves the binomial coefficient formula and simplification.
By rearranging and simplifying the terms as shown in the step-by-step process, we logically conclude that \( \binom{n}{n-1} = n\).
Similarly, showing that \( \binom{n}{n} = 1\) uses the same logical steps: applying the definition and simplifying to arrive at 1.
Developing skills in mathematical proofs can greatly enhance problem-solving abilities and ensure understanding of fundamental concepts.