Pascal's Triangle Calculator & Generator

Generate Pascal's Triangle row by row and explore binomial coefficients, row sums, patterns, symmetry, the binomial theorem, and other important mathematical properties.

Generate Pascal's Triangle

Enter the number of rows you want to generate. The calculator displays each row of Pascal's Triangle together with its row number.

Enter a number from 1 to 50.

Pascal's Triangle — 10 Rows

n = 0
1
n = 1
1
1
n = 2
1
2
1
n = 3
1
3
3
1
n = 4
1
4
6
4
1
n = 5
1
5
10
10
5
1
n = 6
1
6
15
20
15
6
1
n = 7
1
7
21
35
35
21
7
1
n = 8
1
8
28
56
70
56
28
8
1
n = 9
1
9
36
84
126
126
84
36
9
1

Row Sums

Row \(n\) Row Sum \(2^n\)
0 1 1 1
1 1, 1 2 2
2 1, 2, 1 4 4
3 1, 3, 3, 1 8 8
4 1, 4, 6, 4, 1 16 16
5 1, 5, 10, 10, 5, 1 32 32
6 1, 6, 15, 20, 15, 6, 1 64 64
7 1, 7, 21, 35, 35, 21, 7, 1 128 128
8 1, 8, 28, 56, 70, 56, 28, 8, 1 256 256
9 1, 9, 36, 84, 126, 126, 84, 36, 9, 1 512 512

What Is Pascal's Triangle?

Pascal's Triangle is a triangular arrangement of numbers in which each interior number is obtained by adding the two numbers directly above it. The first few rows are:

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

The numbers in Pascal's Triangle are binomial coefficients. The value in position \(k\) of row \(n\) is:

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

Pascal's Triangle connects combinatorics, algebra, probability, sequences, and the binomial theorem.

How Pascal's Triangle Is Constructed

Every row starts and ends with \(1\). Each number between the two edges is the sum of the two numbers immediately above it.

\[ \binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k} \]

For example, the middle numbers in the fifth row are obtained from the row above:

\[ 1,\;4,\;6,\;4,\;1 \]

\[ 6 = 3 + 3 \] \[ 4 = 1 + 3 \]

Pascal's Triangle and the Binomial Theorem

One of the most important applications of Pascal's Triangle is the expansion of powers of a binomial.

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

For example:

\[ (a+b)^4 = a^4+4a^3b+6a^2b^2+4ab^3+b^4 \]

The coefficients \(1,4,6,4,1\) are exactly the fifth row of Pascal's Triangle.

Important Properties of Pascal's Triangle

Binomial Coefficients Every entry is a binomial coefficient \(\binom{n}{k}\).
Symmetry Each row is symmetric because \(\binom{n}{k}=\binom{n}{n-k}\).
Row Sum The sum of row \(n\) is \(2^n\).
First Diagonal The first diagonal consists entirely of \(1\)'s.
Second Diagonal The second diagonal produces the counting numbers \(1,2,3,4,\ldots\).
Fibonacci Connection Certain shallow diagonal sums produce Fibonacci numbers.

Sum of a Row in Pascal's Triangle

The sum of all entries in row \(n\) has a particularly simple formula:

\[ \sum_{k=0}^{n}\binom{n}{k}=2^n \]

For example, row \(4\) is:

\[ 1+4+6+4+1=16 \] \[ 2^4=16 \]

This identity follows directly from the binomial theorem by setting \(a=1\) and \(b=1\):

\[ (1+1)^n = \sum_{k=0}^{n} \binom{n}{k} \] \[ 2^n = \sum_{k=0}^{n} \binom{n}{k} \]

Pascal's Triangle and Combinations

Pascal's Triangle provides the values of combinations, which count how many ways \(k\) objects can be selected from \(n\) objects when order does not matter.

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

For example:

\[ \binom{5}{2} = \frac{5!}{2!3!} = 10 \]

The value \(10\) appears in row \(5\), at position \(k=2\).

Symmetry in Pascal's Triangle

Every row of Pascal's Triangle reads the same from left to right and right to left. This follows from the identity:

\[ \binom{n}{k} = \binom{n}{n-k} \]

For example, row \(6\) is:

\[ 1,\;6,\;15,\;20,\;15,\;6,\;1 \]

The values at equal distances from the ends are therefore identical.

How to Use the Pascal's Triangle Calculator

  1. Enter the number of rows you want to generate.
  2. Choose a value between 1 and 50.
  3. Click Generate Triangle.
  4. Review the generated rows and binomial coefficients.
  5. For smaller triangles, use the row-sum table to verify the \(2^n\) identity.

The calculator is useful for studying binomial coefficients, combinations, algebraic expansions, number patterns, and introductory combinatorics.

Pascal's Triangle Examples

Example 1: Row 3

\[ 1,\;3,\;3,\;1 \]

The row sum is:

\[ 1+3+3+1=8=2^3 \]

Example 2: Row 5

\[ 1,\;5,\;10,\;10,\;5,\;1 \]

The row sum is:

\[ 1+5+10+10+5+1=32=2^5 \]

Example 3: Expanding a Binomial

Row \(3\) gives the coefficients for:

\[ (a+b)^3 = a^3+3a^2b+3ab^2+b^3 \]

Related Summe Calculators

Explore more calculators for Pascal's Triangle, combinations, summations, sequences, and related mathematical operations.

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Frequently Asked Questions

What is Pascal's Triangle?

Pascal's Triangle is a triangular arrangement of numbers where each interior number is the sum of the two numbers above it. Its entries are binomial coefficients.

What is the formula for Pascal's Triangle?

The entry in row \(n\) and position \(k\) is the binomial coefficient:

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

What is the sum of row \(n\)?

The sum of all numbers in row \(n\) is:

\[ 2^n \]

Why is Pascal's Triangle useful?

It is useful for binomial expansions, combinations, probability, algebra, sequences, number patterns, and combinatorics.

Is Pascal's Triangle symmetric?

Yes. Each row is symmetric because:

\[ \binom{n}{k} = \binom{n}{n-k} \]

What are the first rows of Pascal's Triangle?

Starting with row \(0\), the first rows are:

\[ 1 \] \[ 1,\;1 \] \[ 1,\;2,\;1 \] \[ 1,\;3,\;3,\;1 \] \[ 1,\;4,\;6,\;4,\;1 \]