Pascal’s Triangle
In short: Pascal’s triangle is a triangular array in which each number is the sum of the two numbers directly above it, and row n lists the coefficients of (a + b)ⁿ.
PrecalculusAlgebraA-APR.C.5
What is Pascal's Triangle?
Pascal’s triangle is built from just two rules: put a 1 at the ends of every row, and fill each inside spot with the sum of the two numbers diagonally above it. From that tiny recipe grows a surprising amount of mathematics.
Counting rows from 0, the entries of row \(n\) are the binomial coefficients \(\binom{n}{0}, \binom{n}{1}, \ldots, \binom{n}{n}\), and the entry in position \(k\) is \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\). Each row reads the same forwards and backwards, and the numbers in row \(n\) add up to \(2^n\).
How to spot it
- A problem shows rows of numbers starting 1; 1, 1; 1, 2, 1 and asks for the next row of Pascal’s triangle.
- Coefficients are needed for a small power of a binomial, like the 5th, without a factorial formula.
- A counting question asks how many ways to choose k items from n, and a small table is practical.
- A pattern question mentions the sum of a row or numbers that are symmetric across the middle.
The method
- To draw Pascal’s triangle, write the top row as a single 1 and number the rows starting from 0.
- Start and end each new row with 1.
- Fill each interior entry by adding the two numbers just above it, one to the left and one to the right.
- Read row \(n\) for the coefficients of \((a + b)^n\), and check that the row sums to \(2^n\).
Pascal's Triangle: solved example
Problem
Build row 7 from row 6 (1, 6, 15, 20, 15, 6, 1) and verify its sum.
Solution
Start and end with 1. The inside entries are sums of neighbors: \(1 + 6 = 7\), \(6 + 15 = 21\), \(15 + 20 = 35\), \(20 + 15 = 35\), \(15 + 6 = 21\), \(6 + 1 = 7\).
Row 7 is 1, 7, 21, 35, 35, 21, 7, 1. Its total is \(1 + 7 + 21 + 35 + 35 + 21 + 7 + 1 = 128 = 2^7\).
Try it
Question 1
Row 9 is 1, 9, 36, 84, 126, 126, 84, 36, 9, 1. Write row 10.
Show the answer
1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1.
Question 2
What is the sum of all the numbers in row 11?
Show the answer
\(2^{11} = 2{,}048\).
Pascal's Triangle vs. Binomial Theorem
The array is only a table of numbers produced by adding neighbors. The theorem attaches those numbers to powers of two terms, such as a and b, to expand a bracket raised to a power. One supplies the coefficients; the other explains why they appear in an expansion.
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