Exercise · ZW-COMBINATIONS-WITH-REPETITION-009 · ★★★ Advanced
How many nonnegative integer solutions does \(x_1+x_2+x_3+x_4\le 5\) have?
Choose one option.
Show answer
- A. 126
Explanation
Add a slack variable \(x_5\ge 0\) to get an equality with sum 5: \(\binom{5+5-1}{5}=\binom{9}{5}=126\). Counting only the equality with 4 variables gives 56.
Review the concept: Combinations with repetition
A combination with repetition of size \(k\) from \(n\) types of object is an unordered selection of \(k\) objects in which each type may be chosen several times.
\[\binom{n+k-1}{k}\]
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