The formula to calculate the value of a specific number in Pascal's Triangle is:
\[ C(n, k) = \frac{n!}{k! \times (n-k)!} \]
Where:
Pascal's Triangle is a mathematical concept that presents a triangular array of binomial coefficients. It is named after the French mathematician Blaise Pascal, although other mathematicians had studied it centuries before him in India, Persia, China, and Italy. The triangle is constructed by starting with an apex of 1. Each subsequent row is created by adding the number above and to the left with the number above and to the right, treating blank entries as 0. The rows of Pascal's Triangle are conventionally enumerated, starting with row \( n = 0 \) at the top. 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 used in various areas of mathematics, including algebra, probability theory, and combinatorics.
Let's assume the following values:
Using the formula:
\[ C(5, 2) = \frac{5!}{2! \times (5-2)!} = \frac{120}{2 \times 6} = 10 \]
The value in Pascal's Triangle at row 5 and column 2 is 10.