Question

\( \binom{7}{0}+\binom{7}{1} + \binom{7}{2}+\binom{7}{3} + \cdots + \binom{7}{7} \)

• A. \( 2^8 -2 \)
• B. \( 2^7 -1 \)
• C. \( 2^7 \)
• D. \( 2^8 -1 \)
• E. \( 2^7 -2 \)

Hint:

Sum of all binomial coefficients of order n equals \(2^n\).

Answer 1

The Correct Option is C

Concept: Sum of binomial coefficients.

Step 1: Recall identity.
\[ \sum_{r=0}^{n} \binom{n}{r} = 2^n \]

Step 2: Here \( n=7 \).

Step 3: Substitute: \[ = 2^7 \]

Step 4: Expand meaning: \[ =128 \]

Step 5: Final answer: \[ 2^7 \]