find the coefficient of x^4 in the expansion of (1+x+x^2+x^3)^11 - Maths - Binomial Theorem

Question

find the coefficient of x^4 in the expansion of (1+x+x^2+x^3)^11

Ishwarmani · Accepted Answer

Dear
Student,
Please
find below the solution to the asked query:

As, 1+x+x2+x311= 11+x+x21+x11= 1+x1+x211= 1+x111+x211=1+C111x+C211x2+C311x3+C411x4+C511x5+
1+C111x2+C211x4+C311x6
Since, the coefficient of x4 can be found when 1 is multiplied with C211x4 or C211x2 with C111x2 or C411x4 with 1
So, the coefficient of x4=1×C211+C211×C111+C411×1                                         =1×11!2!×9!+11!2!×9!×11!1!×10!+11!4!×7!×1                                         =11×102+11×102×11+11×10×9×84×3×2                                         =11×102×12+11×10×3                                         =11×10×6+11×10×3                                         =11×10×9                                         =990

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