$1000_{10}$ in binary

1000 is a multiple of 8. Therefore, the digit placeholders representing 1, 2 and 4 should be 0.

Only one of the options have 3 zeroes at the end.

The answer must be 1111101000.

$\begin{aligned} 1000 & = 512 + 488 \\ & = 2^9 + 256 + 232 \\ & = 2^9 + 2^8 + 128 + 104 \\ & = 2^9 + 2^8 +2^7 + 64 + 40 \\ & = 2^9 + 2^8 +2^7 + 2^6 + 32 + 8 \\ & = 2^9 + 2^8 +2^7 + 2^6 + 2^5 + 2^3 \\ & = \boxed{1\ 111\ 101\ 000} \end{aligned}$