In a recent election for class president, Monika received 7 of the 10 votes and Alfred received 3 of the 10 votes that were cast by the class. When the machine was counting the votes, it malfunctioned and instead of giving the vote to the correct person, it gave the vote to each candidate with probability $\frac12$ (regardless of whom the vote was cast for). The probability that the machine gave each student the correct number of votes in the election can be expressed as $a:b$ where $a$ and $b$ are positive, coprime integers. What is the value of $a+b$?
In order for the machine to get the correct outcome, it must award $7$ votes to Monika and $3$ to Alfred. It can do this in any order.
Now use those two numbers to get the desired probability.