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Most Important Selected Qs for JEE AdvancedMathematicsBinomial Theorem

The value of p , for which coefficient of x⁵⁰ in the expression (1+x)¹⁰⁰⁰+2 x(1+x)⁹⁹⁹+3 x^2(1+x)⁹⁹⁸+ .+1001 x¹⁰⁰⁰ is equal to ¹⁰⁰² C_p , is :

Correct answer

50

Step-by-step solution

Co-efficient of x⁵⁰ aligned & S=(1+x)¹⁰⁰⁰+2 x(1+x)⁹⁹⁰+3 x^2(1+x)⁹⁹⁸+ . .+1001 x¹⁰⁰⁰ & x S 1+x =x(1+x)⁹⁹⁹+2 x^2(1+x)⁹⁹⁸ . .+1000 x¹⁰⁰⁰+ 1001 x¹⁰⁰¹ (1+x) aligned aligned & S 1+x =(1+x)¹⁰⁰⁰+x(1+x)⁹⁹⁹+ +x¹⁰⁰⁰- 1001 x¹⁰⁰¹ 1+x & S 1+x =(1+x)¹⁰⁰⁰ [ 1- ( x 1+x )¹⁰⁰¹ 1- x 1+x ]- 1001 x¹⁰⁰¹ (1+x) & S=(1+x)¹⁰⁰²-x¹⁰⁰¹(1+x)-1001 x¹⁰⁰¹ & Co-efficient of x⁵⁰=1002 c₅₀ aligned

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