Sum of a – b + ab, b + c – bc and c – a – ac is

Options

A. `2c + ab – ac – bc`
B. `2c – ab – ac – bc`
C. `2c + ab + ac + bc`
D. `2c – ab + ac + bc`

Answer

To find the sum of the expressions \( a - b + ab \), \( b + c - bc \), and \( c - a - ac \), we will add them step by step.

### Step 1: Write down the expressions
We have three expressions:
1. \( a - b + ab \)
2. \( b + c - bc \)
3. \( c - a - ac \)

### Step 2: Combine the expressions
Now, we will sum them up:
\[ (a - b + ab) + (b + c - bc) + (c - a - ac) \]

### Step 3: Rearrange the terms
Next, we can rearrange the terms by grouping similar types together:
\[ = a - b + ab + b + c - bc + c - a - ac \]

### Step 4: Combine like terms
Now, we will combine like terms:
- Combine \( a \) and \( -a \): \( a - a = 0 \)
- Combine \( -b \) and \( b \): \( -b + b = 0 \)
- Combine \( c \) and \( c \): \( c + c = 2c \)
- The remaining terms are \( ab - bc - ac \)

Putting it all together, we have:
\[ 0 + 0 + 2c + (ab - bc - ac) = 2c + ab - bc - ac \]

### Final Result
Thus, the sum of the given expressions is:
\[ 2c + ab - bc - ac \]

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