GATE CSE
First time here? Checkout the FAQ!
x
+2 votes
198 views

The coefficient of $x^{3}$ in the expansion of $(1 + x)^{3} (2 + x^{2})^{10}$ is.

  1. $2^{14}$
  2. $31$
  3. $\left ( \frac{3}{3} \right ) + \left ( \frac{10}{1} \right )$
  4. $\left ( \frac{3}{3} \right ) + 2\left ( \frac{10}{1} \right )$
  5. $\left ( \frac{3}{3} \right ) \left ( \frac{10}{1} \right ) 2^{9}$

 

asked in Set Theory & Algebra by Veteran (27.7k points)   | 198 views

3 Answers

+6 votes
Best answer

$(1+x)^3 = (1+x^{3}+3x+3x^{2})$

and $(2+x^{2})^{10} = _{0}^{10}\textrm{C}*2^{0}*(x^{2})^{10} + _{1}^{10}\textrm{C}*2^{1}*(x^{2})^{9} + ................ + _{9}^{10}\textrm{C}*2^{9}*(x^{2})^{1} + _{10}^{10}\textrm{C}*2^{10}*(x^{2})^{0}$

 

So , coefficient of $x^{3} = _{10}^{10}\textrm{C} * 2^{10} + 3 * _{9}^{10}\textrm{C}*2^{9} = 2^{9} (32) = 2^{14}$

As here we need to multiply last term of second expansion with first term of first coefficient ( x3 ) and 3x with x2 in the second expansion.

answered by Boss (5.3k points)  
selected by
+3 votes

it should be 2^14

equivallent exp:(1+x)3210(1+x2/2)10 

 we can get x^3 in the expansion if (1+x),(1+x) and (1+x)  is multiplied together or one of the (1+x)  block multiplied with one of the (2+x^2)  block

so,coff of x^3= (3c3 + 3c1*10c1*1/2).*2^10  =  (1+3*10*1/2)*2^10  =2^14

answered by Active (2k points)  
0 votes

(1+x)3 (2+ x2 )10

Lets Expand this ->

(1+x)(1+x) (1+x) (2+ x2 )(2+ x2 )(2+ x2 )(2+ x2 )(2+ x2 )(2+ x2 )(2+ x2 )(2+ x2 )(2+ x2 )(2+ x2 )

If you see here there are two ways of getting x3

Way 0 :->Choose x from each of (1+x)3 In that case we need to choose 2 from each of (2+ x2 ). Here we get 210 part of sum.

Way 1-> Here we can choose (1+x) from (1+x)3 . Which we can do in 3 ways (3 choose 1) . Then we need to choose 1 xfrom (2+ x2 )10 , Which we can do in 10 way ( 10 choose 1)

Total ways we can choose x3 using way 1 is 30 ! (10* 3)

Total sum of coeffient in way 1 = 29 * 30

After calculation we get = 210 + 29 * 30 => 16384 => 214 => Answer A)

answered by Veteran (40.6k points)  
Top Users Jan 2017
  1. Debashish Deka

    7172 Points

  2. Habibkhan

    4696 Points

  3. Vijay Thakur

    4308 Points

  4. sudsho

    4090 Points

  5. saurabh rai

    4024 Points

  6. Arjun

    3292 Points

  7. santhoshdevulapally

    3066 Points

  8. GateSet

    3016 Points

  9. Bikram

    3014 Points

  10. Sushant Gokhale

    2892 Points

Monthly Topper: Rs. 500 gift card

18,838 questions
23,808 answers
51,589 comments
20,148 users