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Q.
If the coefficient of $x^{78}$ in the expansion of $\left(1+x+2 x^2+4 x^4\right)^{20}$ is $\lambda \cdot 2^{40}$ then $\lambda$ is equal to
Binomial Theorem
Solution:
$ \left(1+ x +2 x ^2+4 x ^4\right)^{20} $
$T _{ r +1}= { }^{20} C _{ r } \cdot\left(1+ x +2 x ^2\right)^{20- r } \cdot\left(4 x ^4\right)^{ r } $
$\text { For } r =19$
$ T _{19+1}={ }^{20} C _{19} \cdot\left(1+ x +2 x ^2\right)^1 \cdot\left(4 x ^4\right)^{19}$
$\therefore $ Coefficient of $x^{78}=2 \cdot{ }^{20} C _{10} \cdot 4^{19}=2 \cdot 20 \cdot 2^{38}=10 \cdot 2^{40}$