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Mathematics
The coefficient of x4 in the expansion of (1+x+x2+x3)n, is:
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Q. The coefficient of $ {{x}^{4}} $ in the expansion of $ {{(1+x+{{x}^{2}}+{{x}^{3}})}^{n}}, $ is:
Bihar CECE
Bihar CECE 2002
A
$ {{\,}^{n}}{{C}_{4}} $
B
$ {{\,}^{n}}{{C}_{4}}+{{\,}^{n}}{{C}_{2}} $
C
$ {{\,}^{n}}{{C}_{4}}+{{\,}^{n}}{{C}_{4}}+{{\,}^{n}}{{C}_{2}} $
D
$ {{\,}^{n}}{{C}_{4}}+{{\,}^{n}}{{C}_{2}}^{n}{{C}_{1}}+{{\,}^{n}}{{C}_{2}} $
Solution:
$ {{(1+x+{{x}^{2}}+{{x}^{3}})}^{n}} $
$ ={{[(1+x)+{{x}^{2}}(1+x)]}^{n}} $
$ ={{(1+x)}^{n}}{{(1+{{x}^{2}})}^{n}} $
$ =(1+{{\,}^{n}}{{C}_{1}}x{{+}^{n}}{{C}_{2}}{{x}^{2}}{{+}^{n}}{{C}_{3}}{{x}^{3}}{{+}^{n}}{{C}_{4}}{{x}^{4}}+...) $
$ \times \,(1{{+}^{n}}{{C}_{1}}{{x}^{2}}{{+}^{n}}{{C}_{2}}{{x}^{4}}+...) $
$ \therefore $ Coefficient of $ {{x}^{4}} $
$ ={{\,}^{n}}{{C}_{2}}{{+}^{n}}{{C}_{1}}^{n}{{C}_{2}}{{+}^{n}}{{C}_{4}} $