Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard
  1. Home
  2. Engineering
  3. Solve this problem - Binomial theorem and its simple applications - JEE Main-2
# Engineering

If ^{20}C_{1}+(2^{2}) ^{20}C_{2}+(3^{2}) + ^{20}C_{3} + \cdots \cdots

+(20^{2}) ^{20}C_{20}=A(2^{\beta }), then the ordered pair (A, \beta ) is equal to :

 

  • Option 1)

    (420, 19)

  • Option 2)

    (420, 18)

     

  • Option 3)

    (380, 18)

     

  • Option 4)

    (380, 19)

     

Answers (1)
V Vakul

^{20}C_{1}+(2^{2}) ^{20}C_{2}+(3^{2}) + ^{20}C_{3} + \cdots \cdots+(20^{2}) ^{20}C_{20}=A(2^{\beta })

LHS : 

\sum_{r=1}^{20}r^{2}\cdot (_{20}^{r}C\textrm{})

=>\sum_{r=1}^{20}r\cdot (r\cdot _{20}^{r}C\textrm{})

=>20\sum_{r=1}^{20}(r\cdot _{r-1}^{19}C\textrm{})

=>20\sum_{r=1}^{20}(r-1+1) _{r-1}^{19}C\textrm{}

=>20\sum_{r=1}^{20}(r-1) _{r-1}^{19}C\textrm{}+20\sum_{r=1}^{20}_{r-1}^{19}C\textrm{}

=>20\times 19\sum_{r=2}^{20}_{r-2}^{18}C\textrm{}+20\times2^{19}

=>20\times 19\times2^{18}+20\times2^{19}

=>20\times 2^{18}(19+2)

=>20\times21\times 2^{18}

=>420\times 2^{18}

Comparing LHS with RHS

A=420;\beta =18

 


Option 1)

(420, 19)

Option 2)

(420, 18)

 

Option 3)

(380, 18)

 

Option 4)

(380, 19)

 

Similar Questions

Latest Asked Questions

Most Viewed Questions

Preparation Products

Knockout JEE Main (Six Month Subscription)

- AI Coach Study Modules, - Unlimited Mock Tests, - Study Improvement Plan.

₹ 9999/- ₹ 8499/-
Buy Now
Knockout JEE Main (Nine Month Subscription)

- AI Coach Study Modules, - Unlimited Mock Tests, - Study Improvement Plan.

₹ 13999/- ₹ 12499/-
Buy Now
Test Series JEE Main 2024

Chapter/Subject/Full Mock Tests for JEE Main, Personalized Performance Report, Weakness Sheet, Complete Answer Key,.

₹ 7999/- ₹ 4999/-
Buy Now
Knockout JEE Main (One Month Subscription)

- AI Coach Study Modules, - Unlimited Mock Tests, - Study Improvement Plan.

₹ 7999/- ₹ 4999/-
Buy Now
Knockout JEE Main (Twelve Months Subscription)

- AI Coach Study Modules, - Unlimited Mock Tests, - Study Improvement Plan.

₹ 19999/- ₹ 14499/-
Buy Now
Boost your Preparation for JEE Main 2021 with Personlized Coaching
Start Now
 
Exams
Articles
Questions