Euge said:Hi JGalway,
To prove the result formally, I suggest using the principle of mathematical induction.
Pascal's Triangle is a mathematical tool that can be used to easily calculate the values of $\binom{n}{r}$, also known as combinations. In Pascal's Triangle, each number is the sum of the two numbers directly above it. By counting the number of paths from the top of the triangle to a particular number, we can see that this represents the number of ways to choose r objects from a set of n objects. Therefore, Pascal's Triangle can be used to prove the formula for $\binom{n}{r}$.
The formula for $\binom{n}{r}$ is $\frac{n!}{r!(n-r)!}$, where n is the total number of objects and r is the number of objects being chosen. This formula can also be represented visually with Pascal's Triangle.
In Pascal's Triangle, each row represents the value of $\binom{n}{r}$ for a particular value of n. The numbers in each row are also related to the numbers in the row above it. Specifically, the value of $\binom{n}{r}$ in a row is equal to the sum of the two values directly above it, $\binom{n-1}{r}$ and $\binom{n-1}{r-1}$. This relationship can be seen visually in Pascal's Triangle and can also be proven algebraically.
Yes, Pascal's Triangle can be used to prove other mathematical concepts such as the Binomial Theorem and the Fibonacci sequence. It is a versatile tool that can be used in many different areas of mathematics.
While Pascal's Triangle is a helpful tool for understanding and proving the formula for $\binom{n}{r}$, it is not a comprehensive proof on its own. It is important to also understand the algebraic and combinatorial principles behind the formula in order to fully understand its proof.