Number of integral solutions to the equation \(x+y+z=21\), where \(x \geq 1\), \(y \geq 3\), \(z \geq 4\), is equal to ___
Number of integral solutions to the equation \(x+y+z=21\), where \(x \geq 1\), \(y \geq 3\), \(z \geq 4\), is equal to ___
Show Hint
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:
Correct Answer: 105
Solution and Explanation
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 \(105\).
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Concepts Used:
Integral
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.
F'(x) = f(x)
For every value of x = I.
Types of Integrals:
Integral calculus helps to resolve two major types of problems:
- The problem of getting a function if its derivative is given.
- The problem of getting the area bounded by the graph of a function under given situations.