`(11)_(2)+(22)_(3)+(33)_(4)+(55)_(6)+(66)_(7)+(77)_(8)+(88)_(9)=(?)_(10)`

Options

A. 396
B. 276
C. 250
D. 342

Answer

To solve the given problem, we need to convert each number from its respective base to decimal (base 10) and then sum them up. Finally, we will express the result in base 10.

Let's break down the solution step-by-step:

### Step 1: Convert each number to decimal

1. **Convert (11)₂ to decimal:**
\[ 1 \times 2^1 + 1 \times 2^0 = 1 \times 2 + 1 \times 1 = 2 + 1 = 3 \]

2. **Convert (22)₃ to decimal:**
\[ 2 \times 3^1 + 2 \times 3^0 = 2 \times 3 + 2 \times 1 = 6 + 2 = 8 \]

3. **Convert (33)₄ to decimal:**
\[ 3 \times 4^1 + 3 \times 4^0 = 3 \times 4 + 3 \times 1 = 12 + 3 = 15 \]

4. **Convert (55)₆ to decimal:**
\[ 5 \times 6^1 + 5 \times 6^0 = 5 \times 6 + 5 \times 1 = 30 + 5 = 35 \]

5. **Convert (66)₇ to decimal:**
\[ 6 \times 7^1 + 6 \times 7^0 = 6 \times 7 + 6 \times 1 = 42 + 6 = 48 \]

6. **Convert (77)₈ to decimal:**
\[ 7 \times 8^1 + 7 \times 8^0 = 7 \times 8 + 7 \times 1 = 56 + 7 = 63 \]

7. **Convert (88)₉ to decimal:**
\[ 8 \times 9^1 + 8 \times 9^0 = 8 \times 9 + 8 \times 1 = 72 + 8 = 80 \]

### Step 2: Sum all the decimal values

Now, we will add all the converted decimal values together:
\[ 3 + 8 + 15 + 35 + 48 + 63 + 80 \]

Calculating the sum step-by-step:
- \(3 + 8 = 11\)
- \(11 + 15 = 26\)
- \(26 + 35 = 61\)
- \(61 + 48 = 109\)
- \(109 + 63 = 172\)
- \(172 + 80 = 252\)

### Final Result

Thus, the total sum in decimal (base 10) is:
\[ (11)₂ + (22)₃ + (33)₄ + (55)₆ + (66)₇ + (77)₈ + (88)₉ = (252)_{10} \]

### Answer
The answer is \( (252)_{10} \).

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