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प्रश्न
The coefficient of x5 in the expansion of \[\left( 1 + x \right)^{21} + \left( 1 + x \right)^{22} + . . . + \left( 1 + x \right)^{30}\]
पर्याय
51C5
9C5
31C6 − 21C6
30C5 + 20C5
उत्तर
31C6 − 21C6
\[\text{ We have } \left( 1 + x \right)^{21} + \left( 1 + x \right)^{22} + . . . \left( 1 + x \right)^{30} \]
\[ = \left( 1 + x \right)^{21} \left[ \frac{\left( 1 + x \right)^{10} - 1}{\left( 1 + x \right) - 1} \right]\]
\[ = \frac{1}{x}\left[ \left( 1 + x \right)^{31} - \left( 1 + x \right)^{21} \right]\]
\[\text{ Coefficient of } x^5 \text{ in the given expansion = Coefficient of } x^5 \text{ in } \frac{1}{x}\left[ \left( 1 + x \right)^{31} - \left( 1 + x \right)^{21} \right]\]
\[ = \text{ Coefficient of } x^6 \text{ in }\left[ \left( 1 + x \right)^{31} - \left( 1 + x \right)^{21} \right]\]
\[ =^{31} C_6 -^{21} C_6 \]
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संबंधित प्रश्न
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(iii) \[\left( x - \frac{1}{x} \right)^6\]
\[= ^{5}{}{C}_0 (2x )^5 (3y )^0 +^{5}{}{C}_1 (2x )^4 (3y )^1 + ^{5}{}{C}_2 (2x )^3 (3y )^2 + ^{5}{}{C}_3 (2x )^2 (3y )^3 + ^{5}{}{C}_4 (2x )^1 (3y )^4 +^{5}{}{C}_5 (2x )^0 (3y )^5\]
\[= 32 x^5 + 5 \times 16 x^4 \times 3y + 10 \times 8 x^3 \times 9 y^2 + 10 \times 4 x^2 \times 27 y^3 + 5 \times 2x \times 81 y^4 + 243 y^5 \]
\[ = 32 x^5 + 240 x^4 y + 720 x^3 y^2 + 1080 x^2 y^3 + 810x y^4 + 243 y^5 \]
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