How to Solve Pascal’s Triangle?

Pascal’s triangle can be used for many purposes in mathematics. It is used in probability, to find the number of combinations, in the binomial expansion of a polynomial, and can be used to find the Fibonacci series.

Related Topics

• The Binomial Theorem

A step-by-step guide to Pascal’s triangle

Pascal’s triangle is an arrangement of numbers in a triangular array such that the numbers at the end of each row are \(1\) and the remaining numbers are the sum of the two nearest numbers in the row above. The number of elements in the \(nth\) row is equal to \((n + 1)\) elements.

Pascal’s triangle can be easily constructed by adding the pairs of consecutive numbers in the previous lines and writing them in the new line.

we use Pascal’s triangle formula to fill the number in the \(nth\) column and \(mth\) row of Pascal’s triangle. The formula requires the knowledge of the elements in the \(\left(n-1\right)^{th}\) row, and \(\left(m-1\right)^{th}\) and nth columns. The elements of the \(n^{th}\) row of Pascal’s triangle are given by, \(^nC_0\), \(^nC_1\), \(^nC_2\), …, \(^nC_n\). The formula for Pascal’s triangle is:

\(\color{blue}{^nC_m\:=\:^{n-1}C_{m-1}+\:^{n-1}C_{m}}\)

where

•  \(^nC_m\) showes the \(\left(m+1\right)^{th}\) element in the \(n^{th}\) row.
• \(n\) is a non-negative integer
• \(0≤ m≤n\).

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Pascal’s triangle pattern

Pascal’s triangle has various patterns within the triangle that were discovered and explained by Pascal himself or known before him. Some of Pascal’s triangle patterns are:

• The sum of values in the \(n^{th}\) row is \(2^n\).
• If a row has a second element that is a prime, then all the following elements in the row are divisible by that prime (not including \(1s\)). ex. \(1 5 10 10 5 1\).
• By adding the diagonal elements of Pascal’s triangle, we get the Fibonacci series.

Pascal’s Triangle – Example 1:

Find the \(3rd\) element in the \(4th\) row.

Solution:

This means you want to calculate \(^4C_2\:\). Then according to the formula:

\(^4C_2\:=^{4-1}C_{2-1}+^{4-1}C_2\:\:\)

\(^4C_2=^3C_1+^3C_2\)

So, this means we need to add the \(2nd\) element in the \(3rd\) row (i.e. \(3\)) with the \(3rd\) element in the \(3rd\) row (i.e. \(3\).). So that would be our answer:

\(^4C_2=3+3=6\)

Exercises for Pascal’s Triangle

Solve.

1. Write the \(6th\) row of Pascal’s Triangle.
2. What is the sum of the \(12th\) row of Pascal’s triangle?

1. \(\color{blue}{1, 6, 15, 20, 15, 6, 1}\)
2. \(\color{blue}{4096}\)