Pascals Triangle and the Binomial Theorem Explained

Pascal's triangle is a triangular array of binomial coefficients found in probability theory, combinatorics, and algebra. Pascal’s triangle binomial theorem helps us to calculate the expansion of ${{(a+b)}^{n}}$, which is very difficult to calculate otherwise. Pascal's Triangle is used in a variety of fields, including architecture, graphic design, banking, and mapping.

History of Blaise Pascal

Blaise Pascal

Born: 19th June 1623

Died: 19th August 1662

Nationality: French

Contribution: He gave Pascal's Triangle Binomial Theorem to the world.

Statement of the Pascal's Triangle Binomial Theorem

The Pascal’s Triangle Binomial Theorem states that the formula to find the entry of an element in the ${{n}^{th}}$ row and ${{k}^{th}}$ column of a Pascal’s Triangle is given by

$\left( \begin{array}{*{35}{l}}n \\k \\\end{array} \right)$

The formula listed below can be used to determine the components of the following rows and columns:

$\left( \begin{array}{*{35}{l}} n \\k \\\end{array} \right)=\left( \begin{array}{*{35}{l}} n-1 \\k-1 \\\end{array} \right)+\left( \begin{matrix}n-1 \\k \\\end{matrix} \right)$

Proof of the Pascal's Triangle Binomial Theorem

Assuming\[k\le n\],

$\left( \begin{matrix}n-1 \\k-1 \\\end{matrix} \right)+\left( \begin{matrix}n-1 \\k \\\end{matrix} \right)=\frac{(n-1)!}{(k-1)!(n-k)!}+\frac{(n-1)!}{k!(n-k-1)!}$

$=(n-1)!\left( \frac{k}{k!(n-k)!}+\frac{n-k}{k!(n-k)!} \right)$

$=(n-1)!\cdot \frac{n}{k!(n-k)!}$

$=\frac{n!}{k!(n-k)!}$

$=\left( \begin{array}{*{35}{l}} n \\k \\\end{array} \right)$

Hence proved.

Application of the Pascal's Triangle Binomial Theorem

• We can write big binomial expansions with the help of Pascal’s triangle and we won’t have to use the formulas for binomial expansion.

• Concepts of Pascal’s Triangle are also used in so many Maths puzzles.

Limitations of the Theorem

• For positive integral exponents, the theorem can’t be applied to commutative algebra.

• For negative or fractional exponents, the corresponding infinite series are subject to convergence criteria.

Solved Examples

1. Expand and verify ${{\left( a+b \right)}^{2}}$.

Ans:

Pascal’s Triangle

First, let’s write the generic equation without the coefficients:

${{(a+b)}^{2}}={{c}_{0}}{{a}^{2}}{{b}^{0}}+{{c}_{1}}{{a}^{1}}{{b}^{1}}+{{c}_{2}}{{a}^{0}}{{b}^{2}}$

The last row of the triangle gives us the value of coefficients:

${{c}_{0}}=1$, ${{c}_{1}}=2$, ${{c}_{2}}=1$

Now, let’s write the expansion with coefficients:

${{a}^{2}}+2ab+{{b}^{2}}$

2. Expand and verify ${{(a+b)}^{4}}$.

Ans:

Pascal's Triangle (II)

First, let’s write the generic equation without the coefficients:

${{(a+b)}^{4}}={{c}_{0}}{{a}^{4}}{{b}^{0}}+{{c}_{1}}{{a}^{3}}{{b}^{1}}+{{c}_{2}}{{a}^{2}}{{b}^{2}}+{{c}_{3}}{{a}^{1}}{{b}^{3}}+{{c}_{4}}{{a}^{0}}{{b}^{4}}$

The last row of the triangle gives us the value of coefficients:

${{c}_{0}}=1,{{c}_{1}}=4,{{c}_{2}}=6,{{c}_{3}}=4\text{ }\!\!~\!\!\text{ and }\!\!~\!\!\text{ }{{c}_{4}}=1$

Now let’s write the expansion with coefficients:

${{(a+b)}^{4}}={{a}^{4}}{{b}^{0}}+4{{a}^{3}}{{b}^{1}}+6{{a}^{2}}{{b}^{2}}+4{{a}^{1}}{{b}^{3}}+{{a}^{0}}{{b}^{4}}$

3. Expand ${{(a+b)}^{5}}$.

Ans:

Pascal’s Triangle III

First, let’s write the generic equation without the coefficients:

${{(a+b)}^{5}}={{c}_{0}}{{a}^{5}}{{b}^{0}}+{{c}_{1}}{{a}^{4}}{{b}^{1}}+{{c}_{2}}{{a}^{3}}{{b}^{2}}+{{c}_{3}}{{a}^{2}}{{b}^{3}}+{{c}_{4}}{{a}^{1}}{{b}^{4}}+{{c}_{5}}{{a}^{0}}{{b}^{5}}$

The last row of the triangle gives us the value of coefficients:

${{c}_{0}}=1,{{c}_{1}}=5,{{c}_{2}}=10,{{c}_{3}}=10,{{c}_{4}}=5\text{ }\!\!~\!\!\text{ and }\!\!~\!\!\text{ }{{c}_{5}}=1.$

Now let’s write the expansion with coefficients:

${{(a+b)}^{5}}={{a}^{5}}{{b}^{0}}+5{{a}^{4}}{{b}^{1}}+10{{a}^{3}}{{b}^{2}}+10{{a}^{2}}{{b}^{3}}+5{{a}^{1}}{{b}^{4}}+{{a}^{0}}{{b}^{5}}$

Conclusion

Pascal's Triangle gives us the coefficients for an expanded binomial of the form ${{(a+b)}^{n}}$, where $n$ is the row of the triangle. The Binomial Theorem tells us we can use these coefficients to find the entire expanded binomial, with a couple of extra tricks thrown in.

Important Points to Remember

• Each row's integers represent binomial coefficients.

• The figures on the second diagonal are composed of counting figures.

• The third diagonal numbers are triangular numbers.

• Tetrahedral numbers make up the fourth diagonal numbers.

• Pentatone numbers make up the fifth diagonal.

• The sum of the numbers on each row is a power of 2.