jmc

By the uniqueness of the binary representation of positive integers, there is only one way to represent 400 as a sum of distinct powers of $2$. To find this representation, we convert 400 to binary form. The largest power of $2$ less than 400 is $2^8=256$. The difference between 400 and 256 is 144. The largest power of 2 less than 144 is $2^7=128$. The difference between 144 and 128 is 16. Since $16=2^4$, we have found that $400=2^8+2^7+2^4$. The sum of the exponents of 2 in this representation is $\boxed{19}$.