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Pascal's Triangle Calculator

Enter the number of rows and hit the Calculate button for binomial expansion using Pascal’s triangle calculator.

Triangle Binomial Expansion

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The Pascal triangle calculator constructs the Pascal triangle by using the binomial expansion method. You can save a lot of time by using Pascal’s triangle expansion calculator to quickly build the triangle of numbers at one click.

Let’s go through the binomial expansion equation, method to use Pascal’s triangle without Pascal’s triangle binomial expansion calculator, and few examples to properly understand the technique of making Pascal triangle.

What is Pascal’s triangle?

The triangular array of binomial coefficients is known as Pascal’s triangle. It has higher decimal generalizations. In Pascal’s triangle, each number is the sum of diagonal numbers above it.

Pascal triangle binomial expansion formula

Pascal’s triangle calculator uses the below formula for binomial expansion:

(x+y)2=k=0n(nk)xnkyk(x + y)^2 = \displaystyle\sum_{k=0}^n \enspace \dbinom n k \enspace x^{n-k} y^k

Where,

(nk)=(n1k1)+(n1k)\dbinom n k = \dbinom {n-1} {k-1} + \dbinom {n-1} {k}

Pascal's triangle patterns

Below are the important Pascal triangle patterns:

How to use Pascal’s triangle for binomial expansion?

If you want to learn the method of binomial expansion using Pascal’s triangle, take a look at the below triangle carefully.

Example: Expand (a + b) 4 with Pascal triangle.

Step 1: Write down and simplify the expression if needed.

 (a + b) 4

Step 2: Choose the number of row from the Pascal triangle to expand the expression with coefficients. Because (a + b) 4 has the power of 4, we will go for the row starting with 1, 4. The row starting with 1, 4 is 1 4 6 4 1.

(a + b) 4 = 1 4 6 4 1

Step 3: Use the numbers in that row of the Pascal triangle as coefficients of a and b. Attach a with 1st digit of the row and b with the last digit of the row. All other digits in the row will be associated with ab.

(a + b) 4 = 1a + 4ab + 6ab + 4ab + 1b

Step 4: Place the powers to the variables a and b. Power of a should go from 4 to 0 and power of b should go from 0 to 4.

(a + b) 4 = 1a4 + 4a3b + 6a2b2 + 4ab3+ 1b4

(a + b) 4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4

Hurray! You have done the binomial expansion of (a + b) 4 using the Pascal triangle. You can also get the final expansion at one click if you expand using Pascal’s triangle calculator.

References:

  1. What is Pascal's Triangle by mathsisfun.com
  2. Pascal's Triangle: Definition, Calculating Combinations - statisticshowto.com