What is `(1111)_(2) + (1001)_(2) - (1010)_(2)` equal to?

Options

A. `(111)_(2)`
B. `(1100)_(2)`
C. `(1110)_(2)`
D. `(1010)_(2)`

Answer

To solve the problem `(1111)_(2) + (1001)_(2) - (1010)_(2)`, we will follow these steps:

### Step 1: Convert binary numbers to decimal
1. **Convert (1111)₂ to decimal:**
- \(1 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 1 \times 2^0\)
- \(= 1 \times 8 + 1 \times 4 + 1 \times 2 + 1 \times 1\)
- \(= 8 + 4 + 2 + 1 = 15\)

2. **Convert (1001)₂ to decimal:**
- \(1 \times 2^3 + 0 \times 2^2 + 0 \times 2^1 + 1 \times 2^0\)
- \(= 1 \times 8 + 0 \times 4 + 0 \times 2 + 1 \times 1\)
- \(= 8 + 0 + 0 + 1 = 9\)

3. **Convert (1010)₂ to decimal:**
- \(1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 0 \times 2^0\)
- \(= 1 \times 8 + 0 \times 4 + 1 \times 2 + 0 \times 1\)
- \(= 8 + 0 + 2 + 0 = 10\)

### Step 2: Perform the arithmetic operation in decimal
Now we will calculate \(15 + 9 - 10\):
- \(15 + 9 = 24\)
- \(24 - 10 = 14\)

### Step 3: Convert the result back to binary
Now we need to convert \(14\) back to binary:
1. Divide \(14\) by \(2\):
- Quotient: \(7\), Remainder: \(0\)
2. Divide \(7\) by \(2\):
- Quotient: \(3\), Remainder: \(1\)
3. Divide \(3\) by \(2\):
- Quotient: \(1\), Remainder: \(1\)
4. Divide \(1\) by \(2\):
- Quotient: \(0\), Remainder: \(1\)

Now, read the remainders from bottom to top: \(1110\)

### Final Result
Thus, the result of the operation `(1111)_(2) + (1001)_(2) - (1010)_(2)` is equal to **(1110)₂**.

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