Number of integral solutions to the equation \(x+y+z=21\), where \(x \geq 1\), \(y \geq 3\), \(z \geq 4\), is equal to ___
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To find the number of non-negative integral solutions to an equation with constraints, transform the variables to eliminate the constraints and use the formula for combinations:
Let: \[x' = x - 1, \quad y' = y - 3, \quad z' = z - 4,\] where \(x' \geq 0\), \(y' \geq 0\), \(z' \geq 0\). Then: \[x' + y' + z' = 21 - 1 - 3 - 4 = 13.\] The number of non-negative integral solutions to \(x' + y' + z' = 13\) is given by: \[\binom{13 + 3 - 1}{3 - 1} = \binom{15}{2}.\] \[\binom{15}{2} = \frac{15 \times 14}{2} = 105.\] The total number of integral solutions is \(105\).
The representation of the area of a region under a curve is called to be as integral. The actual value of an integral can be acquired (approximately) by drawing rectangles.
The definite integral of a function can be shown as the area of the region bounded by its graph of the given function between two points in the line.
The area of a region is found by splitting it into thin vertical rectangles and applying the lower and the upper limits, the area of the region is summarized.
An integral of a function over an interval on which the integral is described.
Also, F(x) is known to be a Newton-Leibnitz integral or antiderivative or primitive of a function f(x) on an interval I.
