Integers only

Probability Level 3

Find the sum of coefficients of integer powers of x x in the binomial expansion of ( 1 − 2 x ) 50 (1-2\sqrt{x})^{50} .

Source: IIT-JEE mains 2015.
1 2 ( 3 50 − 1 ) \dfrac{1}{2}(3^{50}-1) 1 2 ( 3 50 + 1 ) \dfrac{1}{2}(3^{50}+1) 1 2 ( 2 50 + 1 ) \dfrac{1}{2}(2^{50}+1) 1 2 ( 3 50 ) \dfrac{1}{2}(3^{50}) 1 1

Sabhrant Sachan by Sabhrant Sachan , 21, India

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1 solution

Sabhrant Sachan
May 10, 2016

To solve this Problem , make two equations : ( 1 − 2 x ) 50 = 1 + ( 50 1 ) ( − 2 x ) + ( 50 2 ) ( − 2 x ) 2 + ( 50 3 ) ( − 2 x ) 3 + ⋯ + ( 50 50 ) ( − 2 x ) 50 ( 1 + 2 x ) 50 = 1 + ( 50 1 ) ( 2 x ) + ( 50 2 ) ( 2 x ) 2 + ( 50 3 ) ( 2 x ) 3 + ⋯ + ( 50 50 ) ( 2 x ) 50 Add the two equations , you get ( 1 − 2 x ) 50 + ( 1 + 2 x ) 50 = 2 [ 1 + ( 50 2 ) ( 2 x ) 2 + ( 50 4 ) ( 2 x ) 4 + ⋯ + ( 50 50 ) ( 2 x ) 50 ] ( 1 − 2 x ) 50 + ( 1 + 2 x ) 50 = 2 [ 1 + ( 50 2 ) ( 2 ) 2 x + ( 50 4 ) ( 2 ) 4 x 2 + ⋯ + ( 50 50 ) ( 2 ) 50 x 25 ] Put x=1 ( 1 − 2 1 ) 50 + ( 1 + 2 1 ) 50 A n s = 1 2 ( 1 + 3 50 ) \text{To solve this Problem , make two equations : } \\ (1-2\sqrt{x})^{50}=1+\dbinom{50}{1}(-2\sqrt{x})+\dbinom{50}{2}(-2\sqrt{x})^2+\dbinom{50}{3}(-2\sqrt{x})^3+\cdots+\dbinom{50}{50}(-2\sqrt{x})^{50} \\ (1+2\sqrt{x})^{50}=1+\dbinom{50}{1}(2\sqrt{x})+\dbinom{50}{2}(2\sqrt{x})^2+\dbinom{50}{3}(2\sqrt{x})^3+\cdots+\dbinom{50}{50}(2\sqrt{x})^{50} \\ \text{Add the two equations , you get } \\ (1-2\sqrt{x})^{50}+(1+2\sqrt{x})^{50}=2[1+\dbinom{50}{2}(2\sqrt{x})^2+\dbinom{50}{4}(2\sqrt{x})^4+\cdots+\dbinom{50}{50}(2\sqrt{x})^{50}] \\ (1-2\sqrt{x})^{50}+(1+2\sqrt{x})^{50}=2[1+\dbinom{50}{2}(2)^2x+\dbinom{50}{4}(2)^4x^2+\cdots+\dbinom{50}{50}(2)^{50}x^{25}] \\ \text{Put x=1 } \\(1-2\sqrt{1})^{50}+(1+2\sqrt{1})^{50} \\ \boxed{Ans =\dfrac12(1+3^{50})}

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Question of jee main 2015

Gaurav Chahar - 5 years, 1 month ago

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Thanks for the information :)

Sabhrant Sachan - 5 years, 1 month ago

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