The value of `: [ C ( 7,0) + C ( 7,1) + C ( 7,2) ]+"........."+[C ( 7,6)+C( 7,7)]` is `:`

Options

A. 254
B. 255
C. 256
D. 257

Answer

To solve the problem, we need to find the sum of the binomial coefficients from \( C(7,0) \) to \( C(7,7) \).

### Step-by-step Solution:

1. **Understanding the Binomial Coefficients**:
The expression we are given is:
\[ C(7,0) + C(7,1) + C(7,2) + C(7,3) + C(7,4) + C(7,5) + C(7,6) + C(7,7) \]
These coefficients represent the number of ways to choose \( r \) items from \( n \) items, where \( n = 7 \).

2. **Using the Binomial Theorem**:
According to the Binomial Theorem, the sum of the coefficients in the expansion of \( (a + b)^n \) is given by:
\[ (1 + 1)^n = 2^n \]
For \( n = 7 \):
\[ (1 + 1)^7 = 2^7 = 128 \]

3. **Conclusion**:
Therefore, the sum of all the binomial coefficients from \( C(7,0) \) to \( C(7,7) \) is:
\[ C(7,0) + C(7,1) + C(7,2) + C(7,3) + C(7,4) + C(7,5) + C(7,6) + C(7,7) = 128 \]

### Final Answer:
The value of the expression is \( 128 \).

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