The Formula and Applications of Pascal’s Triangle

Pascal’s Triangle Formula

The formula for determining the value of an element in the nth row and kth column of Pascal’s triangle is as follows:

It is commonly called “n choose k” and written like this:

The above notation for “n choose k” can also be written as C(n,k) or n C_k or _nC _k.

The first row and the first column have zero values. The topmost row of Pascal’s triangle is row “0,” and the leftmost column in the triangle is column “0.”. This phenomenon is shown in the following triangle.

Let us find term 3 in row 4 using the formula:

Pascal’s Triangle Binomial Expansion

Pascal’s Triangle has many interesting properties and applications, one of which is its connection to binomial expansions. The coefficients in the expansion of (a + b)ⁿ can be found in the nth row of Pascal’s triangle.

For example, let’s consider the binomial expansion of (a + b)³:

(a + b)³ = 1a³ + 3a²b + 3ab² + 1b³

If you look at the fourth row of Pascal’s triangle (starting from the “1” at the top), you can see the coefficients 1, 3, 3, 1, which match the coefficients in the expansion.

        1
       1 1
      1 2 1
     1 3 3 1
    1 4 6 4 1
   1 5 10 10 5 1
  1 6 15 20 15 6 1

If you look at the fourth row of Pascal’s triangle (starting from the “1” at the top), you can see the coefficients 1, 3, 3, 1, which match the coefficients in the expansion.

The following is the general formula for the binomial expansion of (a+b)n:

where

is the binomial coefficient, which is the kth number in the nth row of Pascal’s triangle.

Pascal’s Triangle also shows us the coefficients in the binomial expansion:

Power	Binomial Expansion	Pascal’s Triangle
2	(x + 1)2 = 1x2 + 2x + 1	1, 2, 1
3	(x + 1)3 = 1x3 + 3x2 + 3x + 1	1, 3, 3, 1
4	(x + 1)4 = 1x4 + 4x3 + 6x2 + 4x + 1	1, 4, 6, 4, 1

Pascals Triangle Probability

Pascal’s Triangle has applications in various areas of mathematics, including probability and combinatorics. The example of coin tosses illustrates how the coefficients in Pascal’s triangle are connected to the outcomes of a binomial experiment.

When you’re tossing a coin, you have two possible outcomes: heads (H) or tails (T). Each toss is like a Bernoulli trial, and when you perform multiple tosses, you’re dealing with a binomial distribution.

Let’s look at your example for two tosses:

1. HH
2. HT
3. TH
4. TT

The coefficients in the second row of Pascal’s triangle are 1, 2, and 1. These coefficients represent the number of ways you can get 0, 1, or 2 heads (H) in two tosses.

• 1 way to get 0 heads (TT)
• 2 ways to get 1 head (HT or TH)
• One way to get two heads (HH)

The pattern continues for more tosses, and the coefficients in the corresponding row of Pascal’s triangle give you the number of ways to get different outcomes.

This connection between Pascal’s Triangle and combinatorics is not limited to coin tosses; it extends to various scenarios where you’re counting combinations or outcomes in a similar manner. It’s a versatile tool in mathematics with applications in probability, algebra, and number theory.