Yanghui triangle is a number type, also called Pascal's triangle.
The YangHui triangle is a triangular arrangement of numbers that gives the coefficients when any binomial expression unfolds. The numbers are arranged like a triangle. First, 1 is placed at the top, and then we start putting the number in a triangle pattern. The number we get at each step is the addition of the above two numbers.
Yang Hui triangle
Most people know Yang Hui's triangle through a set of seemingly random rules. Starting with the 1 above, there are 1s on both sides of the triangle. Each new number is located in two numbers and below, and its value is the sum of the two numbers above. Theoretically, the triangle is infinite and extends downward forever, but only the first 6 lines appear in Figure 1 below.
Construction of Yang Hui triangles
The easiest way to construct a triangle is to start with line 0 and write only the number 1. From here, to get the following lines of numbers, add the numbers directly above and directly to the right. If there are no numbers on the left or right, replace a zero for the missing number and proceed with addition. This is the illustration of lines 0 through 5.
From the image above, if we look diagonally, the first diagonal line is a list of 1s, the second diagonal line is a list of counted numbers, the third diagonal line is a list of triangle numbers, and so on.
How to use the YangHui triangle?
Yanghui triangles can be used for a variety of probabilistic conditions. Suppose we toss a coin once, then there are only two possible outcomes, positive (H) or reverse (T).
If you throw twice, there is one possibility that both sides are positive HH, and both sides are negative TT, but at least there are two possibilities for the front or tail, that is, HT or TH.
Now you can consider how yanghui triangles help you.
Let's take a look at the table given here based on the number of throws and the results.
<col>
Number of throws
The number of results
Yang Hui Triangle
1
H
T
1,1
2
HH
HT TH
TT
1, 2, 1
3
HHH
HHT, HTH, THH
HTT, THT, TTH
TTT
1,3,3,1
We can also expand by increasing the number of throws.
The shape of Yang Hui's triangle
1) Line addition: An interesting property of this triangle is that the sum of a row of numbers is equal
where n is the line number:
1 = 1 =
1 + 1 = 2 =
1 + 2 + 1 = 4 =
1 + 3 + 3 + 1 = 8 =
1 + 4 + 6 + 4 + 1 = 16 =
2) Prime numbers in triangles: Another visible pattern in the triangle deals with prime numbers. If a line is a prime row, all numbers in that row (excluding 1) can be divisible by that prime number. For example, if we look at line 5 (1 5 10 10 5 1), we can see that 5 and 10 are divisible by 5.
3) Fibonacci sequence in the triangle: Add the numbers on the diagonal of Pascal's triangle to get the Fibonacci sequence as shown in the following figure.
Characteristics of Yang Hui's triangle
Each number is the sum of the two numbers above.
The external numbers are all 1.
Triangles are symmetrical.
The first diagonal line indicates the number being counted.
The sum of the rows is a power of 2.
Each row is a multiplier of 11, e.g. the fifth line = 14641.
Each number is a "binomial coefficient".
The Fibonacci numbers are listed diagonally.
This is an 18-line Yang Hui triangle;
formula
The formula for finding the element items in column k of row n of Yang Hui's triangle is:
Binomial extended application of Yang Hui's triangle
Yanghui's triangle defines the coefficients that appear in binomial expansions. This means that the nth row of Yang Hui's triangle contains the coefficients of the polynomial expansion expression, which is expanded to:
The coefficient of the unfolded equation above, Cn(r), happens to be the number in the nth row of Yang Hui's triangle. namely:
Let a=b=1, then:
The formula for the dichotomy r+1 term is as follows:
Binomial coefficients have many properties, such as: