Pascal’s Triangle Generator and Coefficient Calculator
Generate Pascal’s triangle up to any row, or calculate a single binomial coefficient using row n and column k.
Generate Pascal’s Triangle
Row and column numbering both start at 0, so the topmost row is row 0.
How Pascal’s triangle is calculated
Pascal’s triangle is calculated by placing a 1 at the top and building each new row from the row above it. Every interior number equals the sum of the two numbers directly above it, while the edges of every row are always 1. This generator builds the triangle using the binomial coefficient formula directly, so it works instantly for any row.
The binomial coefficient formula
The binomial coefficient formula defines every entry in the triangle as a sub n,k equals n factorial divided by the product of k factorial and n minus k factorial. Here n is the row number and k is the column number, both starting from 0, and k can never exceed n.
Pascal’s triangle and binomial expansion
Pascal’s triangle directly provides the coefficients for binomial expansion of x plus y raised to the power n. Row n of the triangle lists the coefficients in order, so row 4 gives 1, 4, 6, 4, 1, which correspond to the terms in the expansion of x plus y to the fourth power.
Example expansion using row 4
Example expansion using row 4 shows how the numbers apply directly. The expansion of x plus y to the fourth power equals x to the fourth, plus 4 x cubed y, plus 6 x squared y squared, plus 4 x y cubed, plus y to the fourth, matching the row 4 values exactly.
Using Pascal’s triangle for combinations and probability
Pascal’s triangle for combinations works because the entry at row n, column k tells you the number of ways to choose k items from a group of n items. For probability problems involving two equally likely outcomes, the sum of row n gives the total number of possible outcomes for n trials.
Coin toss probability example
Coin toss probability example demonstrates this clearly. Tossing a coin three times corresponds to row 3, which sums to 8 total outcomes. The entry at column 1 of row 3 is 3, meaning there are 3 ways to get exactly one heads, giving a probability of 3 divided by 8, or 37.5 percent.
Reference table of Pascal’s triangle rows 0 through 5
| Row | Entries |
|---|---|
| 0 | 1 |
| 1 | 1, 1 |
| 2 | 1, 2, 1 |
| 3 | 1, 3, 3, 1 |
| 4 | 1, 4, 6, 4, 1 |
| 5 | 1, 5, 10, 10, 5, 1 |
Frequently asked questions
Pascal’s triangle is a triangular arrangement of numbers where each number equals the sum of the two numbers directly above it. Row numbering starts at 0, and each row contains the binomial coefficients for that row number.
The formula for any entry in Pascal’s triangle is a sub n,k equals n factorial divided by k factorial times n minus k factorial. This is the same as the binomial coefficient n choose k, written as C(n,k).
To find a number in Pascal’s triangle, identify the row number n and the column number k, then calculate n factorial divided by the product of k factorial and n minus k factorial. Row and column numbering both start at 0.
In binomial expansion of x plus y raised to the power n, the coefficients of each term match the numbers in row n of Pascal’s triangle. For example, row 4 gives 1, 4, 6, 4, 1, which are the coefficients for the x to the y to the 4th power expansion.
Pascal’s triangle starts at row 0 because the top single entry represents zero choose zero, which equals 1. This numbering convention keeps it consistent with binomial coefficient notation, where n choose k requires k to be less than or equal to n starting from zero.
Pascal’s triangle contains several patterns. Each row is symmetrical around its center, the sum of each row equals two times the sum of the row before it, and the second number in each row always matches that row’s number.