Pascals Triangle Calculator
Free Pascals triangle Calculator for number theory. Enter values to get step-by-step solutions with formulas and graphs.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
C(n, k) = n! / (k! x (n-k)!)
Where n is the row number and k is the position within the row (both starting from 0). Each entry equals the number of ways to choose k items from n items, and is the sum of the two entries above it in the triangle.
Worked Examples
Example 1: Binomial Expansion Coefficients
Problem:Find the coefficients for expanding (x + y)^6 using Pascal's Triangle.
Solution:Row 6 of Pascal's Triangle gives the coefficients:\nC(6,0)=1, C(6,1)=6, C(6,2)=15, C(6,3)=20, C(6,4)=15, C(6,5)=6, C(6,6)=1\n(x+y)^6 = x^6 + 6x^5y + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6xy^5 + y^6\nRow sum = 1+6+15+20+15+6+1 = 64 = 2^6
Result:Coefficients: [1, 6, 15, 20, 15, 6, 1] | Row sum: 64
Example 2: Counting Combinations
Problem:How many ways can you choose 3 toppings from 8 available pizza toppings?
Solution:This is C(8, 3) from Pascal's Triangle row 8, position 3:\nC(8,3) = 8! / (3! x 5!) = (8 x 7 x 6) / (3 x 2 x 1) = 336 / 6 = 56\nRow 8: [1, 8, 28, 56, 70, 56, 28, 8, 1]
Result:C(8,3) = 56 ways to choose 3 toppings from 8
Frequently Asked Questions
What is Pascal's Triangle and how is it constructed?
Pascal's Triangle is a triangular array of numbers where each number is the sum of the two numbers directly above it. The triangle starts with a single 1 at the top (row 0), and each subsequent row begins and ends with 1. The interior entries are computed by adding adjacent entries from the previous row. For example, row 4 is [1, 4, 6, 4, 1] because 4 = 1+3, 6 = 3+3, and so on. Named after Blaise Pascal who studied it in the 1600s, the triangle was actually known centuries earlier by Chinese, Persian, and Indian mathematicians. It contains an extraordinary number of mathematical patterns and relationships.
How do binomial coefficients relate to Pascal's Triangle?
Each entry in Pascal's Triangle is a binomial coefficient, written C(n, k) or 'n choose k,' representing the number of ways to choose k items from a set of n items without regard to order. The entry in row n and position k (both starting from 0) equals n factorial divided by (k factorial times (n-k) factorial). These coefficients are the terms in the expansion of (a + b) raised to the nth power, which is why they are called binomial coefficients. For instance, (a + b)^4 = 1a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + 1b^4, matching row 4 of the triangle.
What patterns can be found in Pascal's Triangle?
Pascal's Triangle contains a remarkable number of hidden patterns. The first diagonal is all ones, the second diagonal contains natural numbers (1, 2, 3, 4...), and the third diagonal contains triangular numbers (1, 3, 6, 10...). The sum of each row equals a power of 2: row n sums to 2^n. Coloring even and odd entries differently produces the Sierpinski Triangle fractal. The Fibonacci sequence appears by summing the shallow diagonals running from upper-right to lower-left. The hockey stick pattern states that a diagonal sequence of entries sums to the entry diagonally below the last one. Powers of 11 appear when reading rows as single numbers for small rows.
What is the binomial theorem and how does Pascal's Triangle help?
The binomial theorem states that (a + b)^n equals the sum from k=0 to n of C(n,k) times a^(n-k) times b^k, where C(n,k) is the binomial coefficient. Pascal's Triangle provides these coefficients directly. For example, to expand (x + 2)^3, take row 3 of Pascal's Triangle [1, 3, 3, 1] and write: 1(x^3)(2^0) + 3(x^2)(2^1) + 3(x^1)(2^2) + 1(x^0)(2^3) = x^3 + 6x^2 + 12x + 8. This avoids tedious repeated multiplication and is fundamental in algebra, probability, and combinatorics for expanding polynomial expressions.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy