If we are to have (two, for example) binary numbers, such that their sum is $100111010_2$, and given that the first number has 5 ones, and the second number has 3 ones, can I find the numbers that when added together gave $100111010_2$ (the two numbers are $11010110_2$ and $01100100_2$, by the way)?
My theory is that, the number of partitions of a binary number $n_2$ into $k$ binary parts with the given constraint of ones present in each, there can at most be $k$ ways in which $n_2$ can be partitioned (without order, not necessarily distinct). Anyone here to prove (or disprove) this?