2

Question:

What are the uses of Pascal's Triangle? What are some interesting properties of Pascal's Triangle?

I know that Pascal's Triangle has many uses, but I only know a few of them. I know that the binomial theorem is based on Pascal's Triangle, and I know that the $r$'th element of the $n$'th row is determined by: $$_{n}C_{r}=\binom{n}{r}=\frac{n!}{r!(n-r)!}$$ I also know some interesting properties of Pascal's Triangle. I know that the sum of the elements in the $n$'th row is determined by $2^n$, and I know that if there is a prime number in any row, all the other numbers (except $1$) will be divisible by that prime number.

What other uses / properties of Pascal's Triangle are there?

Anonymous Computer
  • 5,639
  • 1
    I'm not sure of the actual "uses" of the triangle, other than that it visualizes the binomial coefficients in a neat way. And for expanding e.g. $(x + 1)^5$ it can be useful to know the first few rows of the triangle by heart. – TMM Mar 16 '14 at 22:50

2 Answers2

3

if you draw pascal triangle you may see the rows sum $2^n$ then :
$$\sum^n_{k=0} {n \choose k}=2^n $$

exist another property very interesting called stick hockey property enter image description here

if you express this number binomial coefficients you might see for example

$$ \sum^{5}_{k=1}{k+3\choose k-1}={9\choose 4}=126 $$

Pablo Herrera
  • 2,155
0

Fibonacci sequence Try this: make a pattern by going up and then along, then add up the values (as illustrated) ... you will get the Fibonacci sequence. (The Fibonacci sequence starts "0, 1" and then continues by adding the two previous numbers, for example 3+5=8, then 5+8=13, etc.)

alekhya
  • 1