Another way to count the number of nonnegative integral solutions to an equation of the
Chapter 9, Problem 7(choose chapter or problem)
Another way to count the number of nonnegative integral solutions to an equation of the form \(x_{1}+x_{2}+· · ·+x_{n} =m\) is to reduce the problem to one of finding the number of ntuples (\(y_{1}, y_{2}, . . . , y_{n}\)) with \(0 ≤ y_{1} ≤ y_{2} ≤ · · · ≤ y_{n} ≤ m\). The reduction results from letting \(y_{i} = x_{1} + x_{2} +· · ·+ x_{i}\) for each i = 1, 2, . . . , n. Use this approach to derive a general formula for the number of nonnegative integral solutions to \(x_{1} + x_{2} +· · ·+ x_{n} = m\).
Text Transcription:
x_1+x_2+· · ·+x_n =m
y_1, y_2, . . . , y_n
0 ≤ y_1 ≤ y_2 ≤ · · · ≤ y_n ≤ m
y_i = x_1 + x_2 +· · ·+ x_i
x_1 + x_2 +· · ·+ x_n = m
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