CBSE BoardEnglish MediumSTD 11 ScienceMathsBinomial Theorem3 Marks
Question
Using binomial theorem, write down the expansions of the following:
$(2x + 3y)^5$
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Answer
The expansion of $(x + y)^n$ has $n + 1$ term so, the expansion of $(2x + 3y)^5$ has $6$ terms.
Using binomial theorem, we have
$(2\text{x}+3\text{y})^5={^5\text{C}}(2\text{x})^5(3\text{y})^0+{^5\text{C}}_1(2\text{x})^4(3\text{y})^1+{^5\text{C}}_2(2\text{x})^3(3\text{y})^2$
$+{^5\text{C}}_3(2\text{x})^2(3\text{y})^3+{^5\text{C}}_4(2\text{x})(3\text{y})^4+{^5\text{C}}_5(2\text{x})^0(3\text{y})^5$
$=2^5\text{x}^5+5\times2^4\times3\times\text{x}^4\times\text{y}+10\times2^3\times3^2\times\text{x}^3\times\text{y}^2+10\times2^2$
$\times3^3\times\text{x}^2\times\text{y}^3+5\times2\times3^4\times\text{x}\times\text{y}+3^5\text{y}^5$
$=32\text{x}^5+240\text{x}^4\text{y}+720\text{x}^3\text{y}^2+1080\text{x}^2\text{y}^3+810\text{xy}^4+243\text{y}^5$
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