Pascal’S Triangle Definitions and Examples
Introduction
Pascal’s Triangle is a graphical representation of the binomial coefficients. The triangle has many interesting mathematical properties, including: The sum of the elements in each row is a power of two. The sum of the elements in each column is a factorial. The elements in each diagonal sum to the Fibonacci numbers. In this blog post, we will explore these properties and more with some definitions and examples.
Pascal’s Triangle
Pascal’s triangle is a graphical representation of the binomial coefficients. In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem. Pascal’s triangle is named after Blaise Pascal, who described it in 1653.
The rows of Pascal’s triangle are conventionally enumerated starting with row n = 0 at the top (the 0th row). The entries in each row are numbered from the left beginning with k = 0 and are usually staggered relative to the numbers in the adjacent rows. The triangle may be constructed in the following manner: In row 0 (the topmost row), there is only one number, 1. Each number in subsequent rows is constructed by adding together the number above and to the left with the number above and to the right, treating blank entries as 0. For example, 1 3 3 1 is obtained by adding together 1+0=1, 1+3=4 and 3+1=4 . This construction can be continued for any finite number of rows.
What is Pascal’s Triangle?
Pascal’s Triangle is a mathematical triangle that is traditionally used to calculate probabilities. The triangle is named after Blaise Pascal, a French mathematician who popularized its use.
Each row of Pascal’s Triangle contains the numbers 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10… etc. The first and second row will always be 1. To find the number in any other row and column:
-Find the number above and to the left of where you want your answer
-Add that number to the number above and to the right of where you want your answer
-Your answer will be in the middle
Here are some example calculations using Pascal’s Triangle:
-The probability of flipping a coin and getting heads is 50%. This can be calculated by finding the number in the second row and first column. The number above it is 0 (in the first row and first column), and to its right is 1 (in the second row and second column). So 0 + 1 = 1, which is heads.
-The probability of flipping a coin twice and getting heads both times is 25%. This can be calculated by finding the number in the third row and second column. The numbers above it are 1 (in the second row and first column) and 1 (in the third row and first column), so 1 + 1 = 2. So 2/4 = 25%.
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Pascals Triangle Explained
Pascal’s Triangle is a triangular array of numbers that is traditionally used in mathematics and computer science. The triangle gets its name from the French mathematician Blaise Pascal, who first wrote about it in the 1600s.
Each row of Pascal’s Triangle contains the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … (the natural numbers). The first row has only one number in it (the number 1), because there is only one way to make 0 + 1 = 1. The second row has two numbers in it because there are two ways to make 1 + 1 = 2. The third row has three numbers in it because there are three ways to make 2 + 1 = 3.
The triangle can be extended indefinitely in all directions. Each new row always has one more number than the previous row. So the fourth row has four numbers in it (1+2+1), the fifth row has five numbers in it (1+3+3+1), and so on.
The numbers in each row of Pascal’s Triangle add up to a power of two:
2^0 = 1 (first row)
2^1 = 2 (second row)
2^2 = 4 (third row)
2^3 = 8 (fourth row)
and so on…
Pascal’s Triangle Formula
Pascal’s Triangle is an interesting mathematical concept. It is named after Blaise Pascal, a French mathematician who discovered it in the seventeenth century. The triangle is made up of numbers that are arranged in a specific way. Each number in the triangle is the sum of the two numbers above it.
The first few rows of Pascal’s Triangle look like this:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
As you can see, the pattern is that each number is the sum of the two numbers above it. If you were to continue this pattern, you would eventually get to:
1
2 1
3 3 1
4 6 4 1
5 10 10 5 1
6 15 20 15 6 1
7 21 35 35 21 7 1
8 28 56 70 56 28 8 etc…
Pascal’s Triangle Binomial Expansion
In mathematics, Pascal’s triangle is a triangular array of the binomial coefficients. In much of the Western world, it is named after Blaise Pascal, although other mathematicians studied it centuries before him in India, Persia (Iran), China, Germany, and Italy.
The rows of Pascal’s triangle are conventionally enumerated starting with row n = 0 at the top (the 0th row). The entries in each row are numbered from the left beginning with k = 0 and are usually staggered relative to the numbers in the adjacent rows. The triangle may be constructed in the following manner: In row 0 (the topmost row), there is only one entry 1. Each entry of each subsequent row is constructed by adding the number above and to the left with the number above and to the right, treating blank entries as 0. For example, the initial number in the first (or any) column is 1; one enters 1 directly below it in row 1. To construct an entry for any column of any other row (say r), one looks up r ? 1th row: find two entries that flank an empty position between them; these will be X and Y(in our example they will be 2nd and 3rd elements). Then set Z(r) equal to X+Y(in our example Z_3=2+3=5).
Pascals Triangle Probability
Pascal’s Triangle is a triangular array of numbers that begins with a 1 at the top, and each number inside the triangle is the sum of the two numbers directly above it. The triangle gets its name from French mathematician Blaise Pascal.
The first few rows of Pascal’s Triangle look like this:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
You can see that each number inside the triangle is the sum of the two numbers directly above it. For example, the number 3 in row 4 is the sum of the numbers 1 and 2 in row 3.
The probabilities of certain events happening can be found by looking at Pascal’s Triangle. For example, if you flip a coin three times, there are eight possible outcomes: HHH, HHT, HTH, HTT, THH, THT, TTH, and TTT. If you look at row 4 of Pascal’s Triangle, you’ll see that there are eight total outcomes. So, each outcome has a one-eighth chance (or 0.125) of happening.
Pascal’s Triangle Pattern
Pascal’s triangle is an arrangement of numbers that form a triangular pattern. The first row contains a single number, 1. The second row contains two numbers, 1 and 1. The third row contains three numbers, 1, 2, and 1. This pattern continues for each subsequent row, with the next row containing one more number than the previous row.
The numbers in Pascal’s triangle have a special mathematical property known as the binomial coefficients. These coefficients can be used to calculate probabilities and are also found in many other areas of mathematics.
Conclusion
In conclusion, Pascal’s Triangle is a mathematical concept that can be applied to many different fields and disciplines. It is a tool that can be used to help solve problems and to better understand patterns and relationships. Hopefully this article has given you a better understanding of what Pascal’s Triangle is and how it can be used.
