Binomial Theorem — Part 1: Pascal's triangle and the expansion of $(a+b)^{n}$: visual lesson, CBSE Class 11 Mathematics

Lesson

Binomial theorem, Part 1: Expanding a power of a sum

How can the fifth power of a sum be written out without multiplying five brackets one after another?

In this lesson you will

Read the coefficients of a power of a sum from Pascal's triangle
State and prove the binomial theorem for a positive whole-number power
Expand sums, differences and expressions with fractions
Use the theorem to evaluate and approximate powers of numbers
Find remainders with the theorem

The coefficients form a pattern.
Power $n$	Expansion of $(a + b)^{n}$ by multiplying the brackets out
$0$	$1$
$1$	$a + b$
$2$	$a^{2} + 2ab + b^{2}$
$3$	$a^{3} + 3a^{2}b + 3ab^{2} + b^{3}$
$4$	$a^{4} + 4a^{3}b + 6a^{2}b^{2} + 4ab^{3} + b^{4}$
$5$	$a^{5} + 5a^{4}b + 10a^{3}b^{2} + 10a^{2}b^{3} + 5ab^{4} + b^{5}$

Concept

What you will be able to do

Read coefficients from Pascal's triangle
State and prove the binomial theorem
Expand sums, differences and expressions with fractions
Evaluate and approximate numbers such as $99^{3}$ and $1.03^{6}$
Find remainders with the theorem

Concept

Where the coefficients come from

In $(a+b)^{4}$ every term is made by taking $a$ or $b$ from each of the four brackets. A term with $b$ from exactly $r$ brackets is $a^{4-r}b^{r}$ , and the number of ways to pick those $r$ brackets is its coefficient.

\binom{n}{r} = \dfrac{n!}{r!\,(n-r)!}

The coefficient counts the ways of picking the brackets.
Brackets that give $b$ ( $r$ )	Which brackets, numbered $1$ to $4$	Ways	Term formed
$0$	none	$1$	$a^{4}b^{0}$
$1$	{1} {2} {3} {4}	$4$	$a^{3}b^{1}$
$2$	{1,2} {1,3} {2,3} {1,4} {2,4} {3,4}	$6$	$a^{2}b^{2}$
$3$	{1,2,3} {1,2,4} {1,3,4} {2,3,4}	$4$	$a^{1}b^{3}$
$4$	{1,2,3,4}	$1$	$a^{0}b^{4}$

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