Identify the correct ascending order of the following numbers under A-D represented in different bases:

(A) (11101011)2 (Base - 2 binary number)

(B) (564)8 (Base - 8 Octal number)

(C) (614)10 (Base - 10 decimal number)

(D) (489)16 (Base - 16 hexadecimal number)

Choose the correct answer from the options given below:

The correct answer is A, B, C, D.

Important Points 

To convert the numbers to base 10 (decimal), we can use the following formulas:

For binary to decimal:

• (a0 × 2^0) + (a1 × 2^1) + (a2 × 2^2) + ... + (an × 2^n),
• where ai is the i^th digit from the right.

For octal to decimal:

• (a0 × 8^0) + (a1 × 8^1) + (a2 × 8^2) + ... + (an × 8^n),
• where ai is the ith digit from the right.

For hexadecimal to decimal:

• (a0 × 16^0) + (a1 × 16^1) + (a2 × 16^2) + ... + (an × 16^n),
• where ai is the ith digit from the right.
• Letters A-F represents decimal values 10-15.

Key Points

Using these formulas, we can convert the given numbers to decimal form and then arrange them in ascending order:​

So in this case, we first convert each number to decimal:

• (11101011)₂ = (1 × 2^0) + (1 × 2^1) + (0 × 2^2) + (1 × 2^3) + (0 × 2^4) + (1 × 2^5) + (1 × 2^6) + (1 × 2^7) = 235
• (564)₈ = (4 × 8^0) + (6 × 8^1) + (5 × 8^2) = 4 + 48 + 320 = 372
• (614)₁₀ = 614
• (489)₁₆ = (9 × 16^0) + (8 × 16^1) + (4 × 16^2) = 9 + 128 + 1024 = 1161

Then we compare them and arrange them in ascending order:

• (11101011)2< (564)8<(614)10<(489)16

So the correct answer is  A, B, C, D.