The constant term in the expansion of 2x+1x7+3x25 is …..

2x+1x7+3x25=(1+2x8+3x9)5x35 Term independent of x= coefficient of x35 in (1+x8(2+3x))5 5C4x8(3x+2)4= 5C4 coefficient of x3 in (2+3x)4= 5C4×4C3(2)1(3)3=5×4×2×27=1080

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The constant term in the expansion of ${(2 x + \frac{1}{x^{7}} + 3 x^{2})}^{5}$ is …..

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answer is 1080.

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Detailed Solution

${(2 x + \frac{1}{x^{7}} + 3 x^{2})}^{5} = \frac{{(1 + 2 x^{8} + 3 x^{9})}^{5}}{x^{35}}$
$Term independent of x = coefficient of x^{35} in {(1 + x^{8} (2 + 3 x))}^{5}$
$\begin{array}{l} ^{5} C_{4} {(x^{8} (3 x + 2))}^{4} \\ =^{5} C_{4} coefficient of x^{3} in (2 + 3 x)^{4} \\ =^{5} C_{4} \times^{4} C_{3} (2)^{1} (3)^{3} \\ = 5 \times 4 \times 2 \times 27 \\ = 1080 \end{array}$