Question:
The constant term in the expansion of $(1+x)ᵐ(1+\frac1x)ⁿ$ is ________.
The constant term in the expansion of $(1+x)ᵐ(1+\frac1x)ⁿ$ is ________.
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The constant term in the expansion of $(1+x)$ is ____.
Updated On: Apr 15, 2026
- ${}^{m+n}C_{m-1}$
- ${}^{m+n}C_{n}$
- ${}^{m+n}C_{m-n}$
- None of these
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The Correct Option is B
Solution and Explanation
Step 1: Simplify Expression
$(1+x)^m (1 + 1/x)^n = (1+x)^m (\frac{x+1}{x})^n = \frac{(1+x)^{m+n}}{x^n}$.
Step 2: Analysis
Constant term in $\frac{(1+x)^{m+n}}{x^n}$ is the term independent of $x$.
Step 3: Finding the Term
This is equivalent to finding the coefficient of $x^n$ in $(1+x)^{m+n}$.
Step 4: Conclusion
Coefficient $= {}^{m+n}C_{n}$.
Final Answer: (B)
$(1+x)^m (1 + 1/x)^n = (1+x)^m (\frac{x+1}{x})^n = \frac{(1+x)^{m+n}}{x^n}$.
Step 2: Analysis
Constant term in $\frac{(1+x)^{m+n}}{x^n}$ is the term independent of $x$.
Step 3: Finding the Term
This is equivalent to finding the coefficient of $x^n$ in $(1+x)^{m+n}$.
Step 4: Conclusion
Coefficient $= {}^{m+n}C_{n}$.
Final Answer: (B)
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