Show How to Prove $\binom{n}{r}$ with Pascal's Triangle

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In summary, Pascal's Triangle is a mathematical tool that can be used to easily calculate the values of $\binom{n}{r}$, also known as combinations. 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. The formula for $\binom{n}{r}$ is $\frac{n!}{r!(n-r)!}$, and it can also be represented visually with Pascal's Triangle. Additionally, Pascal's Triangle shows the relationship between $\binom{n}{r}$ and $\binom{n-1}{r}$ through the sum of the two numbers directly above it. While it can also be used
  • #1
JGalway
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Repeatedly apply $\binom{n}{r}= \binom{n-1}{r}+\binom{n-1}{r-1}$ to show:

$$\binom{n}{r}=\sum_{i=1}^{r+1}\binom{n-i}{r-i+1}$$

The closest i got was showing you could show different iterations with the binomial coefficients (Pascal's Triangle).
 
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Hi JGalway,

To prove the result formally, I suggest using the principle of mathematical induction.
 
  • #3
Euge said:
Hi JGalway,

To prove the result formally, I suggest using the principle of mathematical induction.

Thanks for the reply.
If it's just induction I think I will just ignore it I thought i was missing some property of ${n \choose r}$.(Never really liked induction)
 

FAQ: Show How to Prove $\binom{n}{r}$ with Pascal's Triangle

How can Pascal's Triangle be used to prove $\binom{n}{r}$?

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 $\binom{n}{r}$.

What is 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.

How does Pascal's Triangle show the relationship between $\binom{n}{r}$ and $\binom{n-1}{r}$?

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.

Can Pascal's Triangle be used to prove other mathematical concepts?

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.

Are there any limitations to using Pascal's Triangle to prove $\binom{n}{r}$?

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.

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