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 $a+b$?

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Probability 12? 12 out of what? Do you mean $\frac{1}{2}$? This looks like it should have the (homework) tag, too. –  Benedict Eastaugh Dec 9 '12 at 13:21
It's okay to ask homework questions but you have to show some effort. –  Karolis Juodelė Dec 9 '12 at 13:22
I think you also want $\frac a b$ instead of $ab$ –  Mark Bennet Dec 9 '12 at 13:50
It seems that the question was copy-pasted without subsequent proofreading and the fraction slashes went missing in the process. –  joriki Dec 9 '12 at 14:01
In that case I've edited it incorrectly and $a:b$ should be $\frac ab$ –  Simon Hayward Dec 9 '12 at 15:12

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.
• There are $\binom{10}3$ ways to choose $3$ of the $10$ votes to be for Alfred, after which the sequence of votes is entirely determined, so there are $\binom{10}3$ sequences of votes that yield the right outcome.
• The machine produces each of the $2^{10}$ possible sequences of votes with probability $\left(\frac12\right)^{10}$.