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A Pascal's triangle is an array of numbers that are arranged in the form of a triangle. It is an equilateral triangle that has a variety of never-ending numbers. The two sides of the triangles have only the number 'one' running all the way down, while the bottom of the triangle is infinite. In algebra, Pascal's triangle gives the coefficients for a binomial expression
The first row of the triangle consists of only the number '1' while the subsequent rows are formed by adding the adjacent numbers in the triangle. For example, if the third row has 2 and 1, the fourth row will have 3 in the center.
Pascal’s Triangle: History
The Pascal's triangle is named after Blaise Pascal, a 17th-century French mathematician. In 1653, the Treatise on the Arithmetical Triangle was written by Pascal, which is today known as Pascal's triangle. Mathematicians first introduced the concept of Pascal’s Triangle in China and Persia, but all of its properties were given by Pascal. Pascal has also given other significant theorems such as foundations of calculus and probability, theorems in geometry, and the Pascaline calculator.
Pascal’s Triangle: Definition
Pascal's triangle can be defined as a triangular array of numbers in which the numbers at the end of the rows are one, and each on the others is the sum of the adjacent numbers.
Pascal’s Triangle: Construction of Pascal's Triangle
- At the top of Pascal's triangle, the number 1 is placed in row zero.
- For the first row, we need to add the adjacent numbers. All the numbers outside the triangle are zero. The following image can explain the 0+1=1 and 0+1=1.
- The exact process will be repeated for all the subsequent rows.
- As this process can be repeated many times, Pascal's triangle thus becomes infinite in nature.
Pascal’s Triangle: Construction of Pascal's Triangle
Pascal’s Triangle: Notation of Pascal's Triangle
The topmost row of Pascal's Triangle is known as the zeroth row, and the next row is known as the first row. According to this convention, each ith row consists of i+1 elements in it. For example, the fourth row will have 4+1= 5 elements.
Pascal’s Triangle: Pascal's Triangle Patterns
- Symmetry- Both sides of Pascal's triangle are identical. The numbers on the right side and the left side are the same.
Symmetry
- Addition of rows- Each of the row in a Pascal’s triangle adds up to give an equal of 2n.
Let us look at given rows to confirm this:
1 = 20
1 + 1 = 2 = 21
1 + 2 + 1 = 4 = 22
1 + 3 + 3 + 1 = 8 = 23
1 + 4 + 6 + 4 + 1 = 16 = 24
1+ 5 + 10 + 10 + 5 + 1 = 32 = 25
- The nth row of Pascal's triangle gives the coefficients of an expanded binomial expression.
For example. The fourth row of Pascal's triangle will contain the coefficients for the binomial expression (x + y)4
(x + y)4 = 1x4 + 4x3y + 4xy3 + 6x2y2 + 1y4
- Hockey stick identity- You can start from any one element in Pascal's triangle, either on the left or right side. Add the sum of the elements in a straight line and stop at any time. The next element diagonally in the opposite direction will be equal to the sum of all numbers.
Hockey stick identity
1+6+21+56=84, which is the next diagonal element in the opposite direction.
Similarly, looking at the red line in the figure
1+12=13, which is the next diagonal element in the opposite direction.
- Exponents of 11- Each line of Pascal's triangle is the power of 11.
110=1
111=11
112=121
113=1331
From the 5th row, the values just overlap each other in this manner.
115=161051
The digits of the fifth row are – 1, 5, 10, 10,5,1.
So,
Exponents of 11
- Sierpinski Gasket- If you color the odd or even elements of Pascal's triangle, you get the same pattern as the Sierpinski Gasket.
Sierpinski Gasket
- Fibonacci Sequence- If you add up the diagonal values in Pascal's triangle, you get the Fibonacci sequence.
Fibonacci Sequence
Pascal’s Triangle: Properties of Pascal's Triangle
- Pascal's triangle is equilateral in nature.
- Both sides only consist of the number 1 and the bottom of the triangle in infinite
- Pascal's triangle has symmetry.
- The sum of every row is given by two raised to the power n.
- Every row gives the digits which are equal to the powers of 11.
- All the numbers that lie outside the triangle are considered to be zero.
Pascal’s Triangle: Formula for finding an element in the triangle
\(\begin{pmatrix} n \\k \end{pmatrix} = \begin{pmatrix} n-1 \\k-1 \end{pmatrix} + \begin{pmatrix} n -1 \\k \end{pmatrix}\)
n the formula, n is the row, and k is the term.
Here, n is non-negative and an integer and 0 ≤ k ≤ n
This notion can also be written as:
\(\begin{pmatrix} n \\k \end{pmatrix} =\frac{n!}{k!(n-k)!}\)
Pascal’s Triangle: Use of Pascal's triangle
- A Pascal's triangle can be used to expand any binomial expression.
For example,
The fourth row of Pascal's triangle will contain the coefficients for the binomial expression (x+y)4
(x + y)4 = 1x4 + 4x3y + 6x2y2 + 4xy3 + 1y4
Use of Pascal's triangle
- Probability- The triangle can be used in various probability conditions. Suppose, we toss a coin once we only have two possibilities heads or tails.
If we toss it twice, we can get HH or TT, but we can also get HT or TH. Now we can see the use of Pascal's triangle.
Tosses | Results | Pascal's triangle |
---|---|---|
1 | H T | 1,1 |
2 | HH HT, TH TT | 1,2, 1 |
3 | HHT, HTH, THH HTT, THT, TTH TTT | 1, 3, 3,1 |
4 | HHHH HHHT, HHTH, HTHH, THHH HHTT, HTHT, HTTH, THHT, THTH, TTHH HTTT, THTT, TTHT, TTTH TTTT | 1, 4, 6, 4, 1 |
Pascal’s Triangle: Frequently Asked Questions (FAQs)
Question: What is the pattern of the diagonals in Pascal's triangle?
Answer. Each diagonal of Pascal's triangle has a specific pattern in it.
- The first diagonal of Pascal's triangle is all ones
- The second diagonal is counting numbers.
- The third diagonal consists of triangular numbers.
- The fourth diagonal consists of tetrahedral numbers.
Pattern of the diagonals in Pascal's triangle
Question: What are the applications of Pascal's Triangle?
Answer. A Pascal's triangle can be used to find the expansion of a binomial expression. The triangle is used to find the probability in some cases, and it also allows us to find the number of combinations possible.
Question: How many different combinations of three can you have if you have a total of 15 balls?
Answer. If we go down to the 15th row of Pascal's triangle and then look at the third value, remembering that the first value must be ignored. We will get the number of combinations possible.
1 14 91 364 ...
1 15 105 455 1365 ...
So, the third value in the 15th sequence is 455. Therefore, there can be 455 combinations if we have a total of 15 balls.
Question: Find the coefficient for the xy+y6 term of the binomial expression (x+y)10
Answer. For finding the coefficient of the expression, we need to look at the sixth term of the 10th line of Pascal's triangle. (remembering we must ignore the first value ).
So the tenth line is- 1 10 45 120 210 252 210 120 45 10 1
So the coefficient will be 210, as it is the sixth value.
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