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Q.
The coefficient of $t^{50}$ in $\left(1 + t^{2}\right)^{25}\left(1 + t^{25}\right)$ $\left(1 + t^{40}\right)\left(1 + t^{45}\right)\left(1 + t^{47}\right)$ is
NTA AbhyasNTA Abhyas 2022
Solution:
$
\left(1+{ }^{25} C_{1} t^{2}+\ldots .+{ }^{25} C_{25} t^{50}\right) \times\left(1+t^{25}+t^{40}+t^{45}+t^{47}\right)
$
As all the terms in the first have even exponent we can ignore $t^{25}, t^{45}$ and $t^{47}$ too thus coefficient of $t^{50}$ is
$
={ }^{25} C_{25}+{ }^{25} C_{5}=1+{ }^{25} C_{5}
$