# How do you use the binomial theorem to expand ${\left( {x + y} \right)^6}$ ?

Answer

Verified

181.5k+ views

**Hint:**In this question, we are given an expression and we have been asked to expand the expression using binomial theorem. Please ensure that you are aware about the binomial theorem formula. Start putting the value of $r$, starting from $0$ and increasing till the value of $n$. Then, simplify all the terms and add them to get a final answer. Also, ensure that you are aware about the formula of combinations.

**Formula used:**1) Binomial Theorem - ${\left( {x + y} \right)^n} = \sum\limits_{r = 0}^n {^n{C_r}{x^{n - r}}{y^r}} $

2) Combinations formula - $^n{C_r} = \dfrac{{n!}}{{\left( {n - r} \right)! \times r!}}$

**Complete step-by-step solution:**

Let us start putting the values of $r$ from $0$ in the formula${\left( {x + y} \right)^n} = \sum\limits_{r = 0}^n {^n{C_r}{x^{n - r}}{y^r}} $.$ \Rightarrow {\left( {x + y} \right)^6} = \sum\limits_{r = 0}^6 {^n{C_r}{x^{n - r}}{y^r}} $

We will first put $r = 0$ then $r = 1$ and take it till $r = 6$. Then, we will add all the terms (As formula requires adding the terms.)

$ \Rightarrow {\left( {x + y} \right)^6}{ = ^6}{C_0}{x^{6 - 0}}{y^0}{ + ^6}{C_1}{x^{6 - 1}}{y^1}{ + ^6}{C_2}{x^{6 - 2}}{y^2}{ + ^6}{C_3}{x^{6 - 3}}{y^3}{ + ^6}{C_4}{x^{6 - 4}}{y^4}{ + ^6}{C_5}{x^{6 - 5}}{y^5}{ + ^6}{C_6}{x^{6 - 6}}{y^6}$Now, we know that if $r + p = n$ then,$^n{C_r}{ = ^n}{C_p}$

Using this identity, we can say that $^6{C_0}{ = ^6}{C_6}$, $^6{C_1}{ = ^6}{C_5}$ and$^6{C_2}{ = ^6}{C_4}$.

Thus, we only have to find the value of $^6{C_0}$, $^6{C_1}$, $^6{C_2}$ and $^6{C_3}$.

Let us first find their values individually.

${ \Rightarrow ^6}{C_0} = \dfrac{{6!}}{{0! \times \left( {6 - 0} \right)!}}$$ = \dfrac{{6!}}{{1 \times 6!}} = 1$ …. $(0! = 1)$

Hence, $^6{C_0}{ = ^6}{C_6} = 1$

${ \Rightarrow ^6}{C_1} = \dfrac{{6!}}{{1! \times \left( {6 - 1} \right)!}} = \dfrac{{6!}}{{5!}} = \dfrac{{6 \times 5!}}{{5!}} = 6$

Hence, $^6{C_1}{ = ^6}{C_5} = 6$

${ \Rightarrow ^6}{C_2} = \dfrac{{6!}}{{\left( {6 - 2} \right)! \times 2!}} = \dfrac{{6!}}{{4! \times 2!}} = \dfrac{{6 \times 5 \times 4!}}{{4! \times 2}} = \dfrac{{30}}{2} = 15$

Hence, $^6{C_2}{ = ^6}{C_4} = 15$

${ \Rightarrow ^6}{C_3} = \dfrac{{6!}}{{3! \times \left( {6 - 3} \right)!}} = \dfrac{{6!}}{{3! \times 3!}} = \dfrac{{6 \times 5 \times 4 \times 3!}}{{3! \times 3 \times 2}} = 20$

Now, we will put these values in the expanded expression.

$ \Rightarrow {\left( {x + y} \right)^6} = {x^6} + 6{x^5}y + 15{x^4}{y^2} + 20{x^3}{y^3} + 15{x^2}{y^4} + 6x{y^5} + {y^6}$

Now, since the powers of every term are different, we cannot simplify them further.

**Hence, our final answer is $ {x^6} + 6{x^5}y + 15{x^4}{y^2} + 20{x^3}{y^3} + 15{x^2}{y^4} + 6x{y^5} + {y^6}$**

**Note:**The two letters in the word – binomial, “bi” means two. A binomial has only two terms. But, when these are multiplied with it, what do we get? To get an answer to this question, we use binomial theorem.

You need not memorize the formula. A trick to write the answer is that give the highest power to x and lowest to y. Then, gradually, with the increase in terms, reduce the powers of x by 1 and increase the powers of y by 1. Continue this till the power of x reaches 0 and power of y reaches maximum.

Recently Updated Pages

If ab and c are unit vectors then left ab2 right+bc2+ca2 class 12 maths JEE_Main

A rod AB of length 4 units moves horizontally when class 11 maths JEE_Main

Evaluate the value of intlimits0pi cos 3xdx A 0 B 1 class 12 maths JEE_Main

Which of the following is correct 1 nleft S cup T right class 10 maths JEE_Main

What is the area of the triangle with vertices Aleft class 11 maths JEE_Main

KCN reacts readily to give a cyanide with A Ethyl alcohol class 12 chemistry JEE_Main

Trending doubts

What was the capital of Kanishka A Mathura B Purushapura class 7 social studies CBSE

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Tropic of Cancer passes through how many states? Name them.

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

Name the Largest and the Smallest Cell in the Human Body ?