Question:
The coefficient of $x^{10}$ in $(1-x^2)(1-x^3)^9$ is:
The coefficient of $x^{10}$ in $(1-x^2)(1-x^3)^9$ is:
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In $(1-x^k)^n$, all powers of $x$ must be multiples of $k$.
Updated On: Apr 28, 2026
- ${}^9C_4$
- $-{}^9C_6$
- $-{}^9C_4$
- ${}^9C_6$
- 0
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The Correct Option is
Solution and Explanation
Step 1: Analysis
Expand $(1-x^3)^9$ using the binomial theorem: $\sum_{r=0}^{9} {}^9C_r (-x^3)^r = \sum {}^9C_r (-1)^r x^{3r}$. The powers of $x$ available are $0, 3, 6, 9, \dots, 27$.
Step 2: Analysis
The total expression is $(1-x^2) \times (\dots + {}^9C_r (-1)^r x^{3r} + \dots)$. To get $x^{10}$, we need: 1. $1 \times x^{10} \implies$ requires $3r = 10$ (No integer solution). 2. $-x^2 \times x^8 \implies$ requires $3r = 8$ (No integer solution).
Step 3: Conclusion
Since no combination results in $x^{10}$, the coefficient is 0. Final Answer: (E)
Expand $(1-x^3)^9$ using the binomial theorem: $\sum_{r=0}^{9} {}^9C_r (-x^3)^r = \sum {}^9C_r (-1)^r x^{3r}$. The powers of $x$ available are $0, 3, 6, 9, \dots, 27$.
Step 2: Analysis
The total expression is $(1-x^2) \times (\dots + {}^9C_r (-1)^r x^{3r} + \dots)$. To get $x^{10}$, we need: 1. $1 \times x^{10} \implies$ requires $3r = 10$ (No integer solution). 2. $-x^2 \times x^8 \implies$ requires $3r = 8$ (No integer solution).
Step 3: Conclusion
Since no combination results in $x^{10}$, the coefficient is 0. Final Answer: (E)
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