Binary of 65535 decimal – UGC NET Question

The number 65535 is equal to 2¹⁶ − 1, which means it is represented in binary as sixteen 1s. The pattern 1111111111111111₂ therefore corresponds exactly to 65535. Any binary string with one fewer one or with a zero somewhere in the middle will give a different decimal result. Hence option B is the correct binary representation.

Option A:
1111111111111110₂ represents 2¹⁶ − 2, since its bits are all 1 except the least significant bit. Its decimal value is 65534, which is one less than 65535. Thus this option is not correct.

Option B:
1111111111111111₂ has ones in all positions from 2⁰ to 2¹⁵, so its value is the sum of all powers of 2 from 1 to 32768. This sum equals 65535, matching the decimal in the question exactly. Therefore this is the correct binary conversion.

Option C:
111111111111111₂ contains only fifteen ones, corresponding to 2¹⁵ − 1 = 32767. This is much smaller than 65535, so it cannot represent the given decimal.

Option D:
1111111111111101₂ is missing a one at the 2¹ position, giving a value of 65533. Since this is less than 65535, this pattern fails to match the number in the question.