Binomial Theorem - Formula, Expansion, Problems and Applications - Practice Questions & MCQ

An algebraic expression consisting of only two terms is called a Binomial Expression.

$
e g \cdot(a+b)^2,\left(\sqrt{x}+\frac{k}{x^2}\right)^5,(x+9 y)^{-2 / 3}
$
If we wanted to expand $(x+y)^{52}$, we might multiply $(x+y)$ by itself fifty-two times. This could take hours!
But if we examine some simple expansions, we can find patterns that will lead us to a shortcut for finding more complicated binomial expansions.

$
\begin{aligned}
& (x+y)^2=x^2+2 x y+y^2 \\
& (x+y)^3=x^3+3 x^2 y+3 x y^2+y^3 \\
& (x+y)^4=x^4+4 x^3 y+6 x^2 y^2+4 x y^3+y^4
\end{aligned}
$
On examining the exponents, we find that with each successive term, the exponent for x decreases by 1 and the exponent for y increases by 1. The sum of the two exponents is n for each term.
Also the coefficients for $(\mathrm{x}+\mathrm{y})^{\mathrm{n}}$ are equal to $\binom{n}{0},\binom{n}{1},\binom{n}{2}, \ldots,\binom{n}{n}$
where, $\binom{n}{r}=C(n, r)={ }^n C_r=\frac{n!}{r!(n-r)!}$
These patterns lead us to the Binomial Theorem, which can be used to expand any binomial expression.
For any natural number n, binomial expansion is

$
\begin{aligned}
& (x+y)^n={ }^n C_0 x^n+{ }^n C_1 x^{n-1} y+{ }^n C_2 x^{n-2} y^2+\cdots+{ }^n C_n y^n \text { where, } n \in \mathbb{N} \\
= & \sum_{r=0}^n\left({ }^n C_r\right) x^{n-r} y^r
\end{aligned}
$
The combination $\binom{n}{r}$ or ${ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}$ is called a binomial coefficient.

Properties of Binomial Theorem and Binomial Coefficient

1. If n is a natural number, then the expansion $(\mathrm{x}+\mathrm{y})^{\mathrm{n}}$ has $\mathrm{n}+1$ terms.
2. In each term, the sum of the index of ' $x$ ' and ' $y$ ' is equal to ' $n$ '.
3. The sum of two consecutive binomial coefficients, ${ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}+{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}-1}={ }^{\mathrm{n}+1} \mathrm{C}_{\mathrm{r}}$. ${ }^n C_x={ }^n C_y$, then either $x=y$ or $n=x+y$
4. ${ }^n C_r={ }^n C_{n-r}=\frac{n!}{r!(n-r)!}$

So, if ${ }^n C_x={ }^n C_y$, then either $x=y$ or $n=x+y$
5. The sum of coefficients $C_0+C_1+C_2+\ldots \ldots \ldots+C_n=2^n$

Properties of Binomial Theorem and Binomial Coefficient

5. Binomial coefficients of the term equidistant from the beginning and end are equal.

Eg, second binomial coefficient from start $={ }^{\mathrm{n}} \mathrm{C}_1$, and second binomial coefficient from the end $={ }^n C_{n-1}$

These are equal as $1+(n-1)=n$
6. ${ }^n C_r=\frac{n}{r} \cdot{ }^{n-1} C_{r-1}$

$
{ }^n C_r=\frac{n!}{r!(n-r)!}=\frac{n \cdot(n-1)!}{r \cdot(r-1)!(n-r)!}=\frac{n}{r} \cdot{ }^{n-1} C_{r-1}
$

7. $\frac{{ }^n C_r}{{ }^n C_{r-1}}=\frac{n-r+1}{r}$