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Mathematics Question on Binomial theorem

Coefficient of t24t^{24} in (1+t2)12)(1+t12(1+t24) (1+t^2)^{12}) (1+t^{12}(1+t^{24}) is

A

12C6+3^{12}C_6 +3

B

12C6+1^{12}C_6 +1

C

12C6^{12}C_6

D

12C6+2^{12}C_6 +2

Answer

12C6+2^{12}C_6 +2

Explanation

Solution

Here, Coefficient of t24t^{24} in (1+t2)12)(1+t12(1+t24)\\{ (1+t^2)^{12}) (1+t^{12}(1+t^{24}) \\}
= Coefficient of t24t^{24} in (1+t2)12(1+t12+t24+t36) \\{(1+t^2)^{12} (1+t^{12}+t^{24}+ t^{36})\\}
= Coefficient of t24t^{24} in
                (1+t2)12+t12(1+t2)12+t24(1+t2)12;\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \\{(1+t^2)^{12}+t^{12}(1+t^2)^{12}+t^{24}(1+t^2)^{12} \\};
\hspace27mm [ neglecting t36(1+t2)12] t^{36}(1 + t^2)^{12}]
= Coefficient of t24 =(12C12+12C6+12C0=2+ 12C6t^{24}\, =(^{12}C_{12}+^{12}C_6+^{12}C_0=2 +\, ^{12}C_6