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$(1 + t^2)^{25} (1 + t^{25}) (1 + t^{40}) (1 + t^{45}) (1 + t^{47})$ ના વિસ્તરણમાં $t^{50}$ નો સહગુણક મેળવો
$1 + ^{25}C_5$
$1 + ^{25}C_5 + ^{25}C_7$
$1 + ^{25}C_7$
એક પણ નહી
Solution
As we are interested in coefficient of $t^{50},$ we shall ignore all the term with exponent more than $50 .$ Thus we can write as
$\left( {1 + {\,^{25}}{{\rm{C}}_1}{{\rm{t}}^2} + \ldots \ldots + {\,^{25}}{{\rm{C}}_{25}}{{\rm{t}}^{50}}} \right) \times \left( {1 + {{\rm{t}}^{25}} + {{\rm{t}}^{40}} + {{\rm{t}}^{45}} + {{\rm{t}}^{47}}} \right)$
As all the terms in the first have even exponent we can ignore $\mathrm{t}^{25}, \mathrm{t}^{45}$ and $\mathrm{t}^{47}$ too thus coefficient of $\mathrm{t}^{50}$ is $ = {\,^{25}}{{\rm{C}}_{25}} + {\,^{25}}{{\rm{C}}_5} = 1 + {\,^{25}}{{\rm{C}}_5}$
