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प्रश्न
Find the coefficient of:
(viii) x in the expansion of \[\left( 1 - 3x + 7 x^2 \right) \left( 1 - x \right)^{16}\]
उत्तर
(viii) Suppose x occurs at the (r + 1)th term in the given expression.
Then, we have:
\[\left( 1 - 3x + 7 x^2 \right) \left( 1 - x \right)^{16} \]
\[ = \left( 1 - 3x + 7 x^2 \right)\left( ^{16}{}{C}_0 + ^{16}{}{C}_1 \left( - x \right) + ^{16}{}{C}_2 \left( - x \right)^2 + ^{16}{}{C}_3 \left( - x \right)^3 + ^{16}{}{C}_4 \left( - x \right)^4 +^{16} C_5 \left( - x \right)^5 + ^{16}{}{C}_6 \left( - x \right)^6 + ^{16}{}{C}_7 \left( - x \right)^7 +^{16}{}{C}_8 \left( - x \right)^8 +^{16}{}{C}_9 \left( - x \right)^9 + ^{16}{}{C}_{10} \left( - x \right)^{10} +^{16}{}{C}_{11} \left( - x \right)^{11} + ^{16}{}{C}_{12} \left( - x \right)^{12} + ^{16}{}{C}_{13} \left( - x \right)^{13} + ^{16}{}{C}_{14} \left( - x \right)^{14} + ^{16}{}{C}_{15} \left( - x \right)^{15} + ^{16}{}{C}_{16} \left( - x \right)^{16} \right)\]
\[ \text{ x occurs in the above expresssion at} {}^{16} C_1 \left( - x \right) - 3x ^{16}{}{C}_0 . \]
\[ \therefore \text{ Coefficient of } x = - \left( \frac{16!}{1! 15!} \right) - 3\left( \frac{16!}{0! 16!} \right) = - 16 - 3 = - 19\]
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संबंधित प्रश्न
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