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There are 5 questions in a quiz designated by:

a, b, c, d, e

Each of the questions can only have one of 4 possible answers:

Possible Answer #1: 5 Possible Answer #2: -5 Possible Answer #3: 7 Possible Answer #4: -7


Am I correct to say that there are 625 possible variations of answers that we can receive?


Can any of the variations equal to zero when we would add all the answers up?

For example on variation

(5) + (-5) + (7) + (-7) + (5) = 5


So the core question is can the sum of all the answers to a particular variation ever be zero.

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Do you mean 5,-5,7, and -7? Also, can you explain the last sentence a little more clearly? – user3180 Feb 4 '11 at 7:57

The answer is no. The sum of a single odd number is odd (since it is just the original number), the sum of two odd numbers is even (see here), and in general, the sum of $n$ odd numbers is odd if $n$ is odd, and even if $n$ is even.

In your problem, you are adding 5 odd numbers, and because 5 is odd, we must have that the sum is odd, regardless of which of the values $5,-5,7,-7$ the numbers $a,b,c,d,e$ take on. However, 0 is an even number. Thus, the sum $a+b+c+d+e$ can never be 0.

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As to the first question ("Am I correct to say that there are 625 possible variations of answers that we can receive?"), if I understand correctly the setting, you have four possible answers for the first question, four for the second and so on. So the number of possible sets of five answers is 4 × 4 × 4 × 4 × 4, i.e. 4^5, i.e 1024.

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