Question:
The sum of the coefficients in the binomial expansion of \( \left( \frac{1}{x} + 2x \right)^6 \) is equal to:
The sum of the coefficients in the binomial expansion of \( \left( \frac{1}{x} + 2x \right)^6 \) is equal to:
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This shortcut works because the expansion is $c_0(x^a) + c_1(x^b) + \dots$. When $x=1$, every term becomes just the coefficient $c_i$, leaving you with their total sum.
Updated On: May 6, 2026
- \( 1024 \)
- \( 729 \)
- \( 243 \)
- \( 512 \)
- \( 64 \)
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The Correct Option is B
Solution and Explanation
Concept:
To find the sum of all coefficients in a polynomial or binomial expansion, substitute every variable in the expression with 1.
Step 1: Substitute \( x = 1 \) into the expression.
\[ \text{Sum of coefficients} = \left( \frac{1}{1} + 2(1) \right)^6 \]
Step 2: Simplify the base.
\[ (1 + 2)^6 = 3^6 \]
Step 3: Calculate the final value.
\[ 3^1 = 3, 3^2 = 9, 3^3 = 27, 3^4 = 81, 3^5 = 243, 3^6 = 729 \]
Step 1: Substitute \( x = 1 \) into the expression.
\[ \text{Sum of coefficients} = \left( \frac{1}{1} + 2(1) \right)^6 \]
Step 2: Simplify the base.
\[ (1 + 2)^6 = 3^6 \]
Step 3: Calculate the final value.
\[ 3^1 = 3, 3^2 = 9, 3^3 = 27, 3^4 = 81, 3^5 = 243, 3^6 = 729 \]
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