Determine the number of integer solutions of x1 + x2 + x3 + x4 + x5 = 32 where xi > 3 for 1 ≤ i ≤ 5
To determine the number of integer solutions, we can use a technique called stars and bars or balls and urns.
In this case, we need to find the number of integer solutions for the equation x1 + x2 + x3 + x4 + x5 = 32, where xi > 3 for 1 ≤ i ≤ 5.
First, let's consider a simpler equation where the xi values can be any non-negative integers (xi ≥ 0). This will help us determine the number of solutions without the constraint xi > 3.
By applying the stars and bars technique, we need to distribute 32 stars (representing the sum of the xi values) among 5 bins (representing the 5 xi variables) with 4 dividers (representing the plus signs in the equation). The number of placements of the dividers determines the solution.
To visualize this, let's represent the stars as "*", the dividers as "|", and the bins as "_":
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In this arrangement, the leftmost bin represents x1, and the rightmost bin represents x5. The number of stars in each bin represents the respective xi values.
Now, let's reintroduce the constraint xi > 3 for 1 ≤ i ≤ 5.
Since each xi value must be greater than 3, we can subtract 4 from each xi to satisfy this condition. This gives us the equation (x1 - 4) + (x2 - 4) + (x3 - 4) + (x4 - 4) + (x5 - 4) = 12, where the new xi values are non-negative.
By applying stars and bars to this modified equation, we need to distribute 12 stars among 5 bins with 4 dividers.
Now, let's visualize the modified equation:
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In this arrangement, each bin represents (xi - 4) for 1 ≤ i ≤ 5. The number of stars in each bin, plus 4, represents the respective xi values.
Using stars and bars, the number of integer solutions to this modified equation is given by (12 + 4 - 1) choose (4) = 15 choose 4 = 1365.
Therefore, there are 1365 integer solutions for x1 + x2 + x3 + x4 + x5 = 32, where xi > 3 for 1 ≤ i ≤ 5.