Pascal's Triangle Row Sum Calculator

Calculate any row of Pascal's Triangle, find its binomial coefficients, and verify that the sum of the row is $2^n$.

What Is Pascal's Triangle?

Pascal's Triangle is a triangular arrangement of numbers in which each row contains binomial coefficients. The first rows are:

$$ \begin{array}{ccccccccc} &&&&1&&&&\\ &&&1&&1&&&\\ &&1&&2&&1&&\\ &1&&3&&3&&1&\\ 1&&4&&6&&4&&1 \end{array} $$

Every interior number is obtained by adding the two numbers directly above it. For example, the $6$ in the fifth row is obtained from $3+3$.

The numbers in row $n$ are the coefficients of the expansion of $(a+b)^n$.

How to Find the Sum of a Pascal's Triangle Row

A fundamental identity of Pascal's Triangle is:

$$ \sum_{k=0}^{n}\binom{n}{k}=2^n $$

This means that once you know the row number $n$, you can find the sum of every number in that row simply by calculating $2^n$.

For example, row 4 is $1,4,6,4,1$. Its sum is $1+4+6+4+1=16$, which equals $2^4$.

Pascal's Triangle and Binomial Coefficients

Combination Formula

$\binom{n}{k}$

The entry in position $k$ of row $n$ is the binomial coefficient:

$$ \binom{n}{k} = \frac{n!}{k!(n-k)!} $$

These coefficients are widely used in combinatorics, probability, algebra, and the binomial theorem.

Binomial Theorem

Expansion of $(a+b)^n$

Pascal's Triangle provides the coefficients used in binomial expansions:

$$ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k}a^{n-k}b^k $$

For example, the coefficients of $(a+b)^4$ are $1,4,6,4,1$.

Properties of Pascal's Triangle

First and last values: Every row begins and ends with 1.
Symmetry: Each row reads the same from left to right and right to left.
Binomial coefficients: The value at position $k$ in row $n$ is $\binom{n}{k}$.
Row sum: The sum of row $n$ is $2^n$.
Adjacent-row relationship: Interior entries are formed by adding adjacent entries from the previous row.
Binomial expansion: The row provides the coefficients for the expansion of $(a+b)^n$.

How to Use This Pascal Row Calculator

Enter a non-negative integer between 0 and 30.
Click Calculate Row & Sum.
The calculator displays every value in the selected row.
The total of the row is displayed separately.
The calculator also shows $2^n$ so you can verify the Pascal row sum identity.

For example, entering 5 produces the row $1,5,10,10,5,1$, whose sum is $32$.

Pascal's Triangle FAQ

What is the sum of row $n$ in Pascal's Triangle?

The sum of all numbers in row $n$, starting with row 0, is $2^n$. For example, row 3 is $1,3,3,1$, and its sum is $8=2^3$.

What is row 0 of Pascal's Triangle?

Row 0 contains only the number $1$. Its sum is therefore $1$, which also agrees with $2^0=1$.

What is the formula for a Pascal's Triangle number?

The number at position $k$ in row $n$ is the binomial coefficient $\binom{n}{k}=\frac{n!}{k!(n-k)!}$.

Why does the sum of a Pascal row equal $2^n$?

The binomial theorem gives $(1+1)^n=\sum_{k=0}^{n}\binom{n}{k}$. Since $1+1=2$, the sum of the coefficients is $2^n$.

What are Pascal's Triangle numbers used for?

Pascal's Triangle is used in binomial expansions, combinations, probability, combinatorics, algebra, and several number-pattern problems.