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Pascal Triangle and Excel COMBIN Function

Blaise Pascal 👶 was born in 1️⃣6️⃣2️⃣3️⃣ and began life as a child prodigy🧑‍🏫. Pascal soon become a celebrated mathematician🧑‍💼.

The Greek mathematician Pythagoras 🧒demonstrated that the square of the longest side of a right triangle, the hypotenuse, is equal to the sum off the squares of the other two sides.c^2=a^2+b^2 This equation has a problem as this will work only for squaring and would not work for any integer higher than 2️⃣ fit the bill. For example, the equation below will not work and provide an expected answer. c^3=a^3+b^3

To address the above problem, Pascal came with a more innovative approach. Pascal used a geometric format to illuminate the underlying algebraic structure. His methodology was simple and is applicable to a wide variety of problems in the world of maths this concept world known as ‘Pascal’s Triangle’🔺. In the Pascal Triangle each number is the sum of the two numbers to the right and to the left on the row above.
1️⃣
1️⃣ 1️⃣
1️⃣ 2️⃣ 1️⃣
1️⃣ 3️⃣ 3️⃣ 1️⃣
1️⃣ 4️⃣ 6️⃣ 4️⃣ 1️⃣
1️⃣ 5️⃣ 1️⃣0️⃣ 1️⃣0️⃣ 5️⃣ 1️⃣
1️⃣ 6️⃣ 1️⃣5️⃣ 2️⃣0️⃣ 1️⃣5️⃣ 6️⃣ 1️⃣

Formula for Pascal’s triangle to find the entry for any row ‘n’ and column ‘k’: This is the same as “n!/ [k! (n – k)!]” Here, n is an integer and 0 ≤ k ≤ n. The formula is also called Pascal’s rule, each entry in the triangle is a binomial coefficient. The COMBIN function in Excel calculates the number of possible combinations when choosing k items from n, without repetition. It can be used to replicate Pascal's Triangle. The screenshot provided displays numbers arranged vertically as row ‘n’ and horizontally as column ‘k’, both starting from zero and increasing by one for each subsequent row and column.
In the COMBIN function, two parameters are required: the first is row ‘n’ [$C9], and the second is column ‘k’ [E$4]. The Excel function ‘IFERROR’ has been used to remove errors from our output. Next, we will attempt to solve the previously mentioned cubic equation using the COMBIN function to calculate the required coefficients. As a solution, we are using the below equation with the help of the COMBIN function calculated co-efficient. Observe that we have leveraged Row [3️⃣] in accordance with Column [0️⃣,1️⃣,2️⃣,3️⃣] from the Pascal Triangle.

(a+b)^3️⃣= 1️⃣ a^3+ 3️⃣ a^2*b+ 3️⃣ a*b^2+ 1️⃣ b^3

We will get 2️⃣1️⃣9️⃣7️⃣ as an answer, now compare this answer =1️⃣3️⃣^3️⃣ with our answer 2️⃣1️⃣9️⃣7️⃣.

It is often noted that Chinese mathematician Chu Shih-chieh invented this in 1️⃣3️⃣0️⃣3️⃣ 📆, before Pascal, using his ‘Precious Mirror of the Four Elements’. Although Pascal acknowledged previous work, he emphasized his novel approach: “Let no one say that I have said nothing new. The arrangement of the subject is new. When we play tennis, we both play with the same ball, but one of us places it better.”

Request everyone try this and do let us know further information required on this. Also refer to attached screenshots for more insight.

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