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Step-by-Step Solution
Step 1: Understand the Problem
We have a rectangle ABCD. The line segments AB, CD, BC, and DA contain 5, 7, 6, and 9 points respectively, in their interiors. We want to calculate:
1. $ \alpha $ = Number of triangles where each vertex is chosen from points on different sides.
2. $ \beta $ = Number of quadrilaterals where each vertex is chosen from points on different sides.
Then find $ \beta - \alpha $.
Step 2: Determine $ \alpha $ (Number of Triangles)
To form a triangle, we must select exactly one point from any three different sides of the rectangle. There are four ways to choose which three sides to pick points from:
Sides CD, BC, DA
Sides AB, BC, DA
Sides AB, CD, DA
Sides AB, CD, BC
Mathematically, using combinations:
$ \alpha = \binom{6}{1}\binom{7}{1}\binom{9}{1} + \binom{5}{1}\binom{7}{1}\binom{9}{1} + \binom{5}{1}\binom{6}{1}\binom{9}{1} + \binom{5}{1}\binom{6}{1}\binom{7}{1} $
Step 3: Calculate $ \alpha $
Compute each term separately:
$ \binom{6}{1}\binom{7}{1}\binom{9}{1} = 6 \times 7 \times 9 = 378 $
$ \binom{5}{1}\binom{7}{1}\binom{9}{1} = 5 \times 7 \times 9 = 315 $
$ \binom{5}{1}\binom{6}{1}\binom{9}{1} = 5 \times 6 \times 9 = 270 $
$ \binom{5}{1}\binom{6}{1}\binom{7}{1} = 5 \times 6 \times 7 = 210 $
Add them all up:
$ \alpha = 378 + 315 + 270 + 210 = 1173
$
Step 4: Determine $ \beta $ (Number of Quadrilaterals)
To form a quadrilateral, we must pick one point from each of the four sides. Thus:
$ \beta = \binom{5}{1}\binom{6}{1}\binom{7}{1}\binom{9}{1} = 5 \times 6 \times 7 \times 9
$
Compute this:
$ \beta = 5 \times 6 \times 7 \times 9 = 1890
$
Step 5: Find $ \beta - \alpha $
Finally, we subtract the number of triangles from the number of quadrilaterals:
$ \beta - \alpha = 1890 - 1173 = 717
$
Answer
The value of $ \beta - \alpha $ is 717.
