Pascal’s triangle refers to an arrangement of numbers in a triangular pattern. It consists of several rows with a single number on the topmost row, then two numbers in the next row, three numbers in the third row, and so on. Thus it forms a triangular structure of numbers. The special characteristic of the triangular pattern is that the numbers are placed in such a way that each number is the sum of two numbers just above the number in the previous row. Pascal’s triangle can be constructed easily by adding the pair of successive numbers in the preceding row and writing them in the new line. Pascal’s triangle was conceptualized by the French mathematician Blaise Pascal. Pascal’s triangle helps to find the coefficients of the terms in case of a binomial expansion in algebra. It also has important applications in statistical calculations.
- Pascal’s triangle is constructed by starting from the single number 1 which is at the topmost row and then 1s are placed at both extreme positions of each row down the triangle until the end.
- The middle numbers are obtained by adding the two numbers just above them. Again, the next row is formed by summing adjacent elements in the previous row.
- The topmost row in Pascal’s triangle is considered to be the 0th row; the next row after that the 1st row, then the 2nd row, and so on.
- In each row, the leftmost element is taken as the 0th element, the element to the right of it is the 1st element, and the next element is considered the 2nd element in that row, and so on.
- So the 1st row contains two elements, the 2nd row has three elements, etc. Therefore, the number of elements in the nth row is equal to (n + 1) elements.
- Pascal’s triangle contains rows in ascending order with 1 as both the extreme left and extreme right element of each row.
- The other elements of a row are obtained by the sum of the two consecutive elements just above it in the previous row.
- Pascal’s triangle formula states the principle to calculate a particular number in any row of Pascal’s triangle. For example, to find the 4th element of the 7th row in Pascal’s triangle, first, we need to know the 3rd element of the 6th row and the 4th element of the 6th row.
- These two numbers are 15 and 20 respectively. By adding these two numbers we get, 15 + 20 = 35, which is the 4th element of the 7th row.
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