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The Pascal's Triangle Calculator generates multiple rows, specific rows or finds individual entries in Pascal's Triangle.

What is Pascal's Triangle

Pascal's triangle is triangular-shaped arrangement of numbers in rows (n) and columns (k) such that each number (a) in a given row and column is calculated as n factorial, divided by k factorial times n minus k factorial. The formula is:

\[ a_{n,k} \equiv \frac{n!}{( k! (n - k)! )} \equiv \binom{n}{k} \]

Note that row and column notation begins with 0 rather than 1. So denoting the number in the first row is a_0,0, the second row is a_1,0, a_1,1, the third row is a_2,0, a_2,1, a_2,2, etc. Also for any single element the column number is less than or equal to its row number, k ≤ n.

What is Pascal's Triangle Used For?

Pascal's triangle is useful in calculating:

• Binomial expansion
• Probability
• Combinatorics

In the binomial expansion of (x + y)ⁿ, the coefficients of each term are the same as the elements of the n^th row in Pascal's triangle. For example if you had (x + y)⁴ the coefficients of each of the xy terms are the same as the numbers in row 4 of the triangle: 1, 4, 6, 4, 1. Keep in mind that where there is no coefficient it's the same as having a coefficient of 1.

(x + y)⁴ = x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴

In probability problems, where there is equal chance of either of two outcomes of an event, the total number of outcomes for n events is the sum of the elements in the n^th row of the triangle.

For example, sum the numbers in the 3^rd row of Pascal's triangle: 1 + 3 + 3 + 1 = 8. So if you were going to toss a coin 3 times in a row, there would be 8 possible outcomes of your sequence of heads/tails: H H H, H H T, H T H, T H H, H T T, T H T, T T H, T T T.

Then you can determine what is the probability that you'd get 1 heads and 2 tails in 3 sequential coin tosses. 3 / 8 = 37.5%.

In combinations problems, Pascal's triangle indicates the number of different ways of choosing k items out of a total of n. You'll find this number in the k^th column of the n^th row of the triangle.

Say you wanted to know how many different ways you could select 2 days out of 5 weekdays. Go to the 5^th row of Pascal's triangle below, and look at the 2^nd column. The 2^nd number in the 5^th row is 10. (Remember, the first row of the triangle is counted as 0, and the first number in any row is counted as 0.)

Combinations Calculator for 2 samples from 5 objects.

Example: Pascal's Triangle Rows 0 through 5

Patterns in Pascal's Triangle

• Each entry is the sum of the two entries above it. Add the upper left and the upper right diagonals to calculate an entry. When an upper diagonal does not exist use 0.
• The second entry and second to last entry in each row is the number of that row (as the first row is row 0). After 0, the row numbers are the natural numbers, counting numbers, or positive integers.
• The entries of the triangle are symmetrical around the center column not including the center column.
• The sum of the entries of any row is two times the the sum of the row preceding it.

For more patterns of Pascal's Triangle see Wikipedia Pascal's Triangle Patterns and Properties.

References

Stover, Christopher and Weisstein, Eric W. "Pascal's Triangle." From MathWorld--A Wolfram Web Resource.