Question:
$1 + {}^{100}C_1 + {}^{100}C_2 + \dots + {}^{100}C_{99} + 1 =$
$1 + {}^{100}C_1 + {}^{100}C_2 + \dots + {}^{100}C_{99} + 1 =$
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Remember ${}^nC_0 = 1$ and ${}^nC_n = 1$ to recognize complete binomial sums.
Updated On: Apr 28, 2026
- $2^{99}$
- $2^{101}$
- $2^{98}$
- $2^{100}$
- $100^2$
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The Correct Option is D
Solution and Explanation
Step 1: Concept
The sum of binomial coefficients is $\sum_{r=0}^{n} {}^nC_r = 2^n$.
Step 2: Analysis
The expression is ${}^{100}C_0 + {}^{100}C_1 + {}^{100}C_2 + \dots + {}^{100}C_{99} + {}^{100}C_{100}$. (Note: $1 = {}^{100}C_0$ and $1 = {}^{100}C_{100}$).
Step 3: Conclusion
This is the complete sum of coefficients for $n = 100$. Sum $= 2^{100}$. Final Answer: (D)
The sum of binomial coefficients is $\sum_{r=0}^{n} {}^nC_r = 2^n$.
Step 2: Analysis
The expression is ${}^{100}C_0 + {}^{100}C_1 + {}^{100}C_2 + \dots + {}^{100}C_{99} + {}^{100}C_{100}$. (Note: $1 = {}^{100}C_0$ and $1 = {}^{100}C_{100}$).
Step 3: Conclusion
This is the complete sum of coefficients for $n = 100$. Sum $= 2^{100}$. Final Answer: (D)
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