Pascal's Triangle Calculator

About Pascal's Triangle Calculator

Pascal's triangle: each number is the sum of the two above it. Row n contains the binomial coefficients C(n,k).

How to use this calculator

1. Choose the row number you want, starting with row 0 at the top.
2. Enter the value of \(n\) for the row you want to generate.
3. Read across the row to see each binomial coefficient \(\binom{n}{k}\).
4. Use the row values for probability, algebra, or pattern practice.

The formula explained

The rule \(\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}\) says each number in Pascal’s triangle is the sum of the two numbers directly above it. These numbers are binomial coefficients, which count combinations.

• n = the row number in Pascal’s triangle
• k = the position of the entry within the row, starting at 0
• \(\binom{n}{k}\) = the binomial coefficient at row n and position k

Step by step method

1. Start with row 0, which is just 1.
2. For each new row, put 1 at both ends.
3. Find every middle number by adding the two numbers above it.
4. Continue until you reach the row you need.

Worked example

Problem. Generate row 4 of Pascal’s triangle.

1. Row 0 is 1.
2. Build the next rows by adding adjacent numbers: row 1 is 1, 1, row 2 is 1, 2, 1, row 3 is 1, 3, 3, 1.
3. Row 4 becomes 1, 4, 6, 4, 1.

Tips and common mistakes

• The first and last numbers in every row are always 1.
• Do not start counting rows at 1 unless the tool specifically says so, because this calculator uses row 0 as the top row.

Frequently asked questions