The value of `(""^7C_0+""^7C_1)+(""^7C_1+""^7C_2)+....(""^7C_6+"^7C_7)` is

Options

A. `2^8-2`
B. `2^8-1`
C. `2^8+1`
D. `2^8`

Answer

To solve the problem, we need to evaluate the expression:

\[ (7C_0 + 7C_1) + (7C_1 + 7C_2) + (7C_2 + 7C_3) + \ldots + (7C_6 + 7C_7) \]

### Step 1: Understand the Structure of the Expression
The expression consists of pairs of binomial coefficients. The first pair is \(7C_0 + 7C_1\), the second is \(7C_1 + 7C_2\), and so on, until the last pair \(7C_6 + 7C_7\).

### Step 2: Combine Like Terms
We can rewrite the entire expression by combining like terms:

\[ 7C_0 + (7C_1 + 7C_1) + (7C_2 + 7C_2) + (7C_3 + 7C_3) + (7C_4 + 7C_4) + (7C_5 + 7C_5) + 7C_6 + 7C_7 \]

This simplifies to:

\[ 7C_0 + 2(7C_1 + 7C_2 + 7C_3 + 7C_4 + 7C_5) + 7C_7 \]

### Step 3: Substitute Values for \(7C_0\) and \(7C_7\)
We know that:

\[ 7C_0 = 1 \quad \text{and} \quad 7C_7 = 1 \]

So, substituting these values gives us:

\[ 1 + 2(7C_1 + 7C_2 + 7C_3 + 7C_4 + 7C_5) + 1 \]

### Step 4: Simplify Further
This can be simplified to:

\[ 2 + 2(7C_1 + 7C_2 + 7C_3 + 7C_4 + 7C_5) \]

### Step 5: Use the Binomial Theorem
The sum of the binomial coefficients from \(7C_0\) to \(7C_7\) is given by the binomial theorem:

\[ \sum_{r=0}^{7} 7C_r = 2^7 = 128 \]

Thus, we can express \(7C_1 + 7C_2 + 7C_3 + 7C_4 + 7C_5\) as:

\[ 7C_1 + 7C_2 + 7C_3 + 7C_4 + 7C_5 = 128 - (7C_0 + 7C_7) = 128 - 2 = 126 \]

### Step 6: Substitute Back into the Expression
Now, substituting back into our expression gives:

\[ 2 + 2 \times 126 = 2 + 252 = 254 \]

### Final Answer
Thus, the value of the expression is:

\[ \boxed{254} \]