If (L) represents a number L in base – M number system, then identify the correct ascending order of the following numbers A-D, when converted to a base-10 number system.

A. (8B)₁₆
B. (212)₈
C. (260)₇
D. (10001101)2
Choose the correct answer from the options given below:

The correct answer is A, B, C, D — meaning when converted to base-10, the values in ascending (smallest to largest) order are A, B, C, D.

To verify this, we convert each number to its base-10 (decimal) equivalent using the standard positional value formula:

$\text{Decimal Value}=\sum _{}{d}_i\times {\mathrm{base}}^i$

where $d_i$ is each digit and i is its positional index starting from 0 on the right.

Step 1 — Convert A: (8B)₁₆ (Hexadecimal, base 16)

In hexadecimal, the letter B represents the decimal value 11. The digits are 8 and B (= 11), from left to right.

${\text{(8B)}}_{16}=8\times {16}^1+11\times {16}^0$

$=8\times 16+11\times 1$

$=128+11=\mathrm{<strong>139</strong>}$

⇒ A = 139 in base 10

Step 2 — Convert B: (212)₈ (Octal, base 8)

The digits are 2, 1, 2 from left to right.

${\text{(212)}}_8=2\times 8^2+1\times 8^1+2\times 8^0$

$=2\times 64+1\times 8+2\times 1$

$=128+8+2=\mathrm{<strong>138</strong>}$

⇒ B = 138 in base 10

Step 3 — Convert C: (260)₇ (Septenary, base 7)

The digits are 2, 6, 0 from left to right.

${\text{(260)}}_7=2\times 7^2+6\times 7^1+0\times 7^0$

$=2\times 49+6\times 7+0$

$=98+42+0=\mathrm{<strong>140</strong>}$

⇒ C = 140 in base 10

Step 4 — Convert D: (10001101)₂ (Binary, base 2)

The 8-bit binary number is: 1 0 0 0 1 1 0 1. Assign positional powers of 2 from right (position 0) to left (position 7):

${\text{(10001101)}}_2=1\times 2^7+0\times 2^6+0\times 2^5+0\times 2^4+1\times 2^3+1\times 2^2+0\times 2^1+1\times 2^0$

$=128+0+0+0+8+4+0+1=\mathrm{<strong>141</strong>}$

⇒ D = 141 in base 10

Step 5 — Summary and Ascending Order

Collecting all converted values:

Arranging in ascending order (smallest → largest):

$138<139<140<141$

Which corresponds to:

$B<A<C<D$

Therefore, the correct ascending order is A, B, C, D, which matches the given correct option.