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  • The coefficient of  in the expansion of <img alt="\left ( 1+x+x^{2} \right )^{10}" src="http://ent

The coefficient of x^{4} in the expansion of \left ( 1+x+x^{2} \right )^{10} is

Option: 1

615


Option: 2

225


Option: 3

450


Option: 4

None 


Answers (1)

best_answer

\left ( 1+x+x^{2} \right )^{10} has general term \frac{10!}{a!\,\,b!\,\,c!}x^bx^{2c}

With a + b + c = 10 .....(i)

For coefficient of x4, b + 2c = 4 .....(ii)

Possible combinations of a, b and c satisfying (i) and (ii) both are

1. When c = 0, b = 4 (from (ii)) , and a = 6 (from (i))

2. When c = 1, b = 2 (from (ii)) , and a = 7 (from (i))

3. When c = 2, b = 0 (from (ii)) , and a = 8 (from (i))

So, (8,0,2), (7,2,1), (6, 4, 0)

So, coefficients are \frac{10!}{8!\,0!\,2!}+ \frac{10!}{7!\,\,2!\,\,1!} + \frac{10!}{6!\,\,4!\,\,0!} = 615

 

Posted by

Divya Prakash Singh

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