Coefficient of $x^{25}$ in $(1 + x + x^{2} + x^{3} + . . . + x^{10})^{7}$ is:

³¹C₁₆ - 7. ²⁰C₁₄

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³¹C₁₄- 7. ²⁰C₁₄

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${}^{31}C_{25}-7.{}^{20}C_{14}+21.{}^{9}C_{3}$

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Solution

The correct option is D
${}^{31}C_{25}-7.{}^{20}C_{14}+21.{}^{9}C_{3}$

$(1 + x + x^{2} + . . . + x^{10})^{7} = (1 - x^{11})^{7} (1 - x)^{- 7}$
= $(1 - 7 x^{11} + 21 x^{22} - 35 x^{33} + . . .) (1 - x)^{- 7}$
coefficient of $x^{25}$ =(coeff. of $x^{25} - 7$ . coeff. of $x^{14} + 21$ .coeff. of $x^{3}$ ) in $(1 - x)^{- 7}$
= $^{31} C_{25} - 7.^{20} C_{14} + 21.^{9} C_{3}$

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