Pascal’s triangle is a triangle formed by numbers by putting a single number on the topmost row, then two numbers in the next row, three numbers in the third row, and so on. So, it forms a triangular arrangement of numbers.
But the most important characteristic of Pascal’s triangle is that the numbers are arranged in such a way that each number in a row is the sum of two numbers just above it in the previous row. The method of construction of Pascal’s triangle involves adding two successive numbers in a row and writing them in the row just below it.
Pascal’s triangle was introduced by the French mathematician Blaise Pascal. Pascal’s triangle has important applications in statistical calculations.
Construction of Pascal’s Triangle
The construction of Pascal’s triangle starts with the number 1 written at the topmost row. The second row has two 1 because of the addition of (0+1) and (1+0). Then 1s are placed at both extreme positions of each row down the triangle until the end.
The middle numbers for each row are obtained as the sum of the two numbers in the preceding row just above them. Again, the next row is formed by adding adjacent numbers in the previous row. The topmost row in Pascal’s triangle is considered to be the 0th row; the next row is the 1st row, then the 2nd row, and so on. In each row, the leftmost element is taken as the 0th element, the next number is the 1st element, and the number next to it is the 2nd element, and so on. It is observed that the 1st row contains two elements; the 2nd row contains three elements, etc. Therefore, the number of elements in the nth row is equal to (n + 1) elements.
To Identify a Number in Pascal’s Triangle
Pascal’s triangle contains rows in ascending order starting from a single element from the top, then two elements in the next row, and so on. It has 1 as both the extreme left and extreme right element of each row. The other elements of a row are obtained by adding the two consecutive elements just above it in the previous row. This particular pattern states the principle that helps to calculate a particular number in any row of Pascal’s triangle. For example, to find the 3rd element of the 5th row in Pascal’s triangle, we need to know the 2nd element of the 4th row and the 3rd element of the 4th row. These two numbers are 4 and 6 respectively. By adding these two numbers we get, 10, which is the 3rd element of the 5th row.
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