The coefficients of (x⁵⁰ ) in ((1+x)¹⁰¹ ) ( (1-x+x^2 )¹⁰⁰ ) is......

Step-by-step solution

Given expression is ( aligned & (1+x)¹⁰¹ (1-x+x^2 )¹⁰⁰ & =(1+x) [(1+x) (1-x+x^2 ) ]¹⁰⁰ & =(1+x) (1+x^3 )¹⁰⁰ aligned ) So, coefficient of (x⁵⁰ ) in ((1+x)¹⁰¹ (1-x+x^2 )¹⁰⁰ ) (= ) coefficient of (x⁵⁰ ) in ((1+x) (1+x^3 )¹⁰⁰ ) (= ) coefficient of (x⁵⁰ ) in ( (1+x^3 )¹⁰⁰ ) + coefficient of (x⁴⁹ ) in ( (1+x^3 )¹⁰⁰ ) ( ) In the expansion of ( (1+x^3 )¹⁰⁰ ), the power of (x ) is multiple of 3 , and while 50 and 49 are not multiple of 3. So, coefficient of (x⁵⁰ ) and 49 in the expansion of ( (1+x^3 )¹⁰⁰ ) is zero. ( ) Coef