The binomial theorem is an important theorem in combinatorics and algebra. It provides a general method for expanding an expression of the form \((x+y)^n\).

 

The formula is based on combinations. Each term in the expansion corresponds to choosing which factors become \(x\), and which become \(y\).

 

 

Example with \( \large n=2 \)

We expand \((x+y)^2\):

 

$$ \large (x+y)^2 = (x+y)(x+y) $$

$$ \large (x+y)^2 = x^2 + xy + yx + y^2 $$

$$ \large (x+y)^2 = x^2 + 2xy + y^2 $$

 

The coefficients in front of each term are \( \large 1,2,1 \).

Because:

• There is 1 \( \large x^2 \)
• There are 2 \( \large xy \)
• There is 1 \( \large y^2 \)

 

 

Example with \( \large n=3 \)

We expand \((x+y)^3\):

 

$$ \large (x+y)^3 = (x+y)(x+y)(x+y) $$

$$ \large (x+y)^3 = x^3 + 3x^2y + 3xy^2 + y^3 $$

 

The coefficients are \( \large 1,3,3,1 \).

 

 

Example with \( \large n=4 \)

We expand \((x+y)^4\):

 

$$ \large (x+y)^4 = (x+y)(x+y)(x+y)(x+y) $$

$$ \large (x+y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4 $$

 

The coefficients are \( \large 1,4,6,4,1 \).

 

 

Pascal’s triangle

The coefficients we obtain can be organized in a triangle called Pascal’s triangle:

 

 

$$ \large \begin{array}{cccccccccccccccccc} & & & & & & & & 1 & & & & & & & \\ & & & & & & & 1 & & 1 & & & & & & \\ & & & & & & 1 & & 2 & & 1 & & & & & \\ & & & & & 1 & & 3 & & 3 & & 1 & & & & \\ & & & & 1 & & 4 & & 6 & & 4 & & 1 & & & \\ & & & 1 & & 5 & & 10 & & 10 & & 5 & & 1 & & \\ & & 1 & & 6 & & 15 & & 20 & & 15 & & 6 & & 1 & \\ & 1 & & 7 & & 21 & & 35 & & 35 & & 21 & & 7 & & 1 \\ 1 & & 8 & & 28 & & 56 & & 70 & & 56 & & 28 & & 8 & & 1 \\ \end{array} $$

 

Each row corresponds to the coefficients in \(\large (x+y)^n\).

 

 

The general formula

The binomial theorem states:

 

$$ \large (x+y)^n = \sum_{r=0}^n \binom{n}{r} x^r y^{n-r} $$

 

Here \(\large \binom{n}{r}\) is a combination, and it indicates how many ways we can choose \(r\) factors of \(x\) out of \(n\).

 

 

Example with \( \large n=5 \)

The expansion of \((x+y)^5\) is:

 

$$ \large (x+y)^5 = x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5 $$

 

The coefficients \( \large 1,5,10,10,5,1 \) correspond to row 5 in Pascal’s triangle.

 

 

Summary

• The binomial theorem provides a method to expand \((x+y)^n\).
• The coefficients are found in Pascal’s triangle.
• The coefficients are given by combinations: \(\large \binom{n}{r}\).

 

The theorem connects algebra, combinatorics and probability, and forms the basis of the binomial distribution.