Binomial Theorem Applications
Easymathematics
The sum C(n,0) - C(n,1) + C(n,2) - C(n,3) + ... equals
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Solution
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- Identity Proof: Consider the expansion of (1+x)n:
- (1+x)n=C(n,0)+C(n,1)x+C(n,2)x2+C(n,3)x3+…
- Substitution: Let x=−1.
- (1−1)n=C(n,0)+C(n,1)(−1)+C(n,2)(−1)2+C(n,3)(−1)3+…
- 0n=C(n,0)−C(n,1)+C(n,2)−C(n,3)+…
- Conclusion: For n≥1, the alternating sum of binomial coefficients is always zero.
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About This Question
- Subject
- mathematics
- Chapter
- binomial theorem and its simple applications
- Topic
- binomial theorem applications
- Difficulty
- Easy
- Year
- 2025
Solution
Correct Answer:
0
- Identity Proof: Consider the expansion of (1+x)n:
- (1+x)n=C(n,0)+C(n,1)x+C(n,2)x2+C(n,3)x3+…
- Substitution: Let x=−1.
- (1−1)n=C(n,0)+C(n,1)(−1)+C(n,2)(−1)2+C(n,3)(−1)3+…
- 0n=C(n,0)−C(n,1)+C(n,2)−C(n,3)+…
- Conclusion: For n≥1, the alternating sum of binomial coefficients is always zero.
This easy difficulty mathematics question is from the chapter binomial theorem and its simple applications, covering the topic of binomial theorem applications. It appeared in the 2025 exam.
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