Download the app

Binomial theorem for positive integral Index

Remember concepts with our Masterclasses.

60 mins Expert Faculty Ask Questions

Question

Let $X = {(^{10} C_{1})}^{2} + 2 {(^{10} C_{2})}^{2} + 3 {(^{10} C_{3})}^{2} + \dots + 10 {(^{10} C_{10})}^{2}$ where $^{10} C_{r}, r \in {1, 2, \dots, 10}$ denote binomial coefficients. Then the value of $\frac{1}{1430} X$ is ____________.

Moderate

Solution

$\begin{matrix} X = \sum_{r = 1}^{10} r \cdot {(^{10} C_{r})}^{2} = \sum_{r = 1}^{10} r \cdot^{10} C_{r} \cdot^{10} C_{r} \\ = 10 \cdot \sum_{r = 1}^{10}^{9} C_{r - 1} \cdot^{10} C_{10 - r} = 10 . cofficient of x^{9} in {(1 + x)}^{9} {(1 + x)}^{10} = 10 . cofficient in {(1 + x)}^{19} = 10 \cdot^{19} C_{9} \end{matrix}$
Now, $\frac{X}{1430} = \frac{10 \cdot^{19} C_{9}}{1430} = \frac{^{19} C_{9}}{143} = \frac{^{19} C_{9}}{11 \times 13}$
$= \frac{19 \times 17 \times 16}{8} = 19 \times 34 = 646$

Ready to Test Your Skills?

Check Your Performance Today with our Free Mock Tests used by Toppers!

Talk to our academic expert!

ctaimg

Create Your Own Test
Your Topic, Your Difficulty, Your Pace

ctaimg

Similar Questions

If the coefficients of P th, ( p + 1)th and (p + 2)th terms in the expansion of (1 + x)ⁿ are in AP, then

799 666 8865

support@infinitylearn.com

6th Floor, NCC Building, Durgamma Cheruvu Road, Vittal Rao Nagar, HITEC City, Hyderabad, Telangana 500081.

free study material

JEE

JEE Mock Tests

JEE study guide

JEE Revision Notes

JEE Important Questions

JEE Sample Papers

JEE Previous Year's Papers

NEET

NEET previous year’s papers

NEET important questions

NEET sample papers

NEET revision notes

NEET study guide

NEET mock tests

CUET

CUET previous year’s papers

CUET important questions

CUET sample papers

CUET revision notes

CUET study guide

CUET mock tests

CBSE

CBSE previous year’s papers

CBSE important questions

CBSE sample papers

CBSE revision notes

CBSE study guide

CBSE mock tests

Popular Books

HC Verma Part 1 HC Verma Part 2 RS Agarwal RD Sharma Lakhmir Singh DK Goel Sandeep Garg TS Grewal

NCERT solutions

Class 12 NCERT Solutions Class 11 NCERT Solutions Class 10 NCERT Solutions Class 9 NCERT Solutions Class 8 NCERT Solutions Class 7 NCERT Solutions Class 6 NCERT Solutions

NCERT exempler

Class 12 NCERT Exemplar Class 11 NCERT Exemplar Class 10 NCERT Exemplar Class 9 NCERT Exemplar Class 8 NCERT Exemplar Class 7 NCERT Exemplar Class 6 NCERT Exemplar

State Board

Rajasthan State Board West Bengal State Board Gujarat State Board Telengana State Board Andhra Pradesh State Board Uttar Pradesh State Board

Subject

Maths Science Physics Chemistry Economics Accounting Business Studies

2023 | www.infinitylearn.com