Answer

Verified

304.5k+ views

**Hint:**We are asked to find the binomial expansion of an expression. For that we should be aware about the general binomial expansion of any expression involving two variables $x$ and $y$ raised to the power of $n$. Moreover, we should be able to find the value of the combination term involved in the binomial expansion.

**Complete step by step answer:**

For any two real numbers $a$ and $b$, and for any natural number $n$, the following binomial expansion holds true:

$\left(a+b\right)^n={}^nC_0a^nb^0+{}^nC_1a^{n-1}b^1+\ldots+{}^nC_na^0b^n$

So, we put $a=2x$, $b=3y$ and $n=5$, we get:

${{\left( 2x+3y \right)}^{5}}{{=}^{5}}{{C}_{0}}{{\left( 2x \right)}^{5}}{{+}^{5}}{{C}_{1}}{{\left( 2x \right)}^{4}}\left( 3y \right){{+}^{5}}{{C}_{2}}{{\left( 2x \right)}^{3}}{{\left( 3y \right)}^{2}}{{+}^{5}}{{C}_{3}}{{\left( 2x \right)}^{2}}{{\left( 3y \right)}^{3}}{{+}^{5}}{{C}_{4}}\left( 2x \right){{\left( 3y \right)}^{4}}{{+}^{5}}{{C}_{5}}{{\left( 3y \right)}^{5}}$

Now, we know that:

${}^nC_r=\dfrac{n!}{r!\left(n!-r!\right)}$

So, after dissolving the coefficients, we have:

${{\left( 2x+3y \right)}^{5}}={{\left( 2x \right)}^{5}}+5{{\left( 2x \right)}^{4}}\left( 3y \right)+10{{\left( 2x \right)}^{3}}{{\left( 3y \right)}^{2}}+10{{\left( 2x \right)}^{2}}{{\left( 3y \right)}^{3}}+5\left( 2x \right){{\left( 3y \right)}^{4}}+{{\left( 3y \right)}^{5}}$

$=32{{\left( x \right)}^{5}}+5\times 16{{\left( x \right)}^{4}}\left( 3y \right)+10{{\left( 2x \right)}^{3}}\times 9{{\left( y \right)}^{2}}+10{{\left( 2x \right)}^{2}}\times 27{{\left( y \right)}^{3}}+5\left( 2x \right)81{{\left( y \right)}^{4}}+243{{\left( y \right)}^{5}}$

$=32{{\left( x \right)}^{5}}+5\times 16{{\left( x \right)}^{4}}\left( 3y \right)+10\times 8{{\left( x \right)}^{3}}\times 9{{\left( y \right)}^{2}}+10\times 4{{\left( x \right)}^{2}}\times 27{{\left( y \right)}^{3}}+5\left( 2x \right)81{{\left( y \right)}^{4}}+243{{\left( y \right)}^{5}}$

$=32x^5+240x^4y+720x^3y^2+1080x^2y^3+810xy^4+243y^5$

Hence, the resultant expression has been obtained.

**Note:**Since the number to which this expression is raised is 5, which is very small; we can simply use the Pascal’s triangle to determine the coefficients of the variables. Using that we would reduce the chances of calculation mistakes that might occur while solving the combination term. The entry in the $n^{th}$ row and $k^{th}$ column of Pascal's triangle is denoted by:

$^{n}{{C}_{k}}$

The first few rows of a Pascal’s triangle are:

Exponent | Coefficients |

N=1 | 1 1 |

N=2 | 1 2 1 |

N=3 | 1 3 3 1 |

N=4 | 1 4 6 4 1 |

N=5 | 1 5 10 10 5 1 |

Using this, we can write the coefficients along with the variables and then do the multiplication to obtain the resultant expression. As we can see this has been obtained only after dissolving the coefficients as we had done in the question.

Recently Updated Pages

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE

Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE

What are the possible quantum number for the last outermost class 11 chemistry CBSE

Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE

What happens when entropy reaches maximum class 11 chemistry JEE_Main

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Write a stanza wise summary of money madness class 11 english CBSE

Which places in India experience sunrise first and class 9 social science CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Difference Between Plant Cell and Animal Cell

Which neighbouring country does not share a boundary class 9 social science CBSE

What is Whales collective noun class 10 english CBSE