Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • Engineering
  • The coefficient of  in <img alt="\left(1+x+x^2+x^3+x^4\right)^n(x-1)^{n+3}" src="https://entrancecorner.o

The coefficient of x^{83} in \left(1+x+x^2+x^3+x^4\right)^n(x-1)^{n+3}, is equal to
 

Option: 1

{ }^n \mathrm{C}_7(-1)^{\mathrm{n}}


Option: 2

-{ }^{\mathrm{n}} \mathrm{C}_{16}


Option: 3

{ }^{n} \mathrm{C}_{13}


Option: 4

{ }^{n} \mathrm{C}_{9}


Answers (1)

best_answer

We have

\begin{aligned} & \left(1+x+x^2+x^3+x^4\right)^r(x-1)^{n+3} \\ & =\left(\frac{1-x^5}{1-x}\right)^n(1-x)^{n+3}=\left(1-x^5\right)^n(1-x)^3 \\ & =\left(-x^3+3 x^2-3 x+1\right) \sum_{r=0}^n{ }^n C_r(-1)^r x^{5 r} \\ & =-\sum_{r=0}^n{ }^n C_r(-1)^r x^{5 r+3}+3 \sum_{r=0}^n{ }^n C_r(-1)^r x^{5 r+2} \\ & \quad-3 \sum_{r=0}^n{ }^n C_r(-1)^r x^{5 r+1}+3 \sum_{r=0}^n{ }^n C_r(-1)^r x^{5 r} \end{aligned}

For term containing \mathrm{x}^{\mathrm{83}}, we have

5 \mathrm{r}+3=83 \Rightarrow \mathrm{r}=16

whereas 5 r+2=83, 5r+1=83 and 5r=83 give no integral value of r.
Hence, there is only one term containing x^{83} whose coefficient

=-{ }^{\mathrm{n}} \mathrm{C}_{16} \text {. }
 

Posted by

Gaurav

View full answer

JEE Main high-scoring chapters and topics

Study 40% syllabus and score up to 100% marks in JEE

Similar Questions

Trending Articles/News

anam-khan

JEE Main 2025: NIT Rourkela cut-offs for CSE, chemical engineering drop; last years' closing ranks
anam-khan

JEE Main 2025 Live: NTA exam city slip at jeemain.nta.nic.in soon; session 1 schedule
anam-khan

JEE Main 2025 schedule out for session 1; paper 1 from January 22

Latest Question