Wondering if I did this question right

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mahjk17

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I am wondering if I did this question right, "How many integer solutions are there to the equation x1 + x2 + x3 + x4 = 32 with
0 <= x (subscript) i <= 10 for i = 1, 2, 3, 4?" My attempt: There is a total of 35 choose 3 answers for all non-negative integers, therefore I must subtract everything that doesn't satisfy the conditions stated in the question which will be anything [x (subscript) i > 10] which will be 25 choose 3. So the answer will be (35 choose 3) - (25 choose 3).. Is that correct?????
 
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mahjk17 said:
I am wondering if I did this question right, "How many integer solutions are there to the equation x1 + x2 + x3 + x4 = 32 with
0 <= x (subscript) i <= 10 for i = 1, 2, 3, 4?" My attempt: There is a total of 35 choose 3 answers for all non-negative integers, therefore I must subtract everything that doesn't satisfy the conditions stated in the question which will be anything [x (subscript) i > 10] which will be 25 choose 3. So the answer will be (35 choose 3) - (25 choose 3).. Is that correct?????
Click to expand...
Look at this expansion. \(\displaystyle {\left( {\sum\limits_{k = 0}^{10} {{x^k}} } \right)^4}\)
The coefficient of \(\displaystyle x^{32}\) is \(\displaystyle 165\). That is the answer.

Now if you must do this by inclusion/exclusion it is difficult.
You have several cases: one variable with more than 10; two variables with more than 10; three variables with more than 10. But at least you know what the answer should be.
 
Thank you very much!! So i should instead use a generating function instead of inclusion exclusion
 
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