Exercise · ZW-COMBINATIONS-WITH-REPETITION-003 · ★★☆ Intermediate
How many nonnegative integer solutions does \(x_1+x_2+x_3=7\) have?
Choose one option.
Show answer
- A. 36
Explanation
It amounts to distributing 7 units among 3 variables: \(\binom{7+3-1}{7}=\binom{9}{7}=36\). The 21 comes from using \(\binom{7}{2}\) and the 15 from the version with positive variables.
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.
\[\binom{n+k-1}{k}\]
More practice problems: combinations with repetition
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- ZW-COMBINATIONS-WITH-REPETITION-004 How many positive integer solutions does x_1+x_2+x_3=7 have?
- ZW-COMBINATIONS-WITH-REPETITION-005 6 objects are chosen from 4 types, with repetition and without order. How many selections are there?
- ZW-COMBINATIONS-WITH-REPETITION-006 2 items are chosen from 4 types, ignoring order. How many more selections are there if repetition i…
- ZW-COMBINATIONS-WITH-REPETITION-007 How many distinct monomials of degree 3 are there in the variables x,y,z?
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- ZW-COMBINATIONS-WITH-REPETITION-009 How many nonnegative integer solutions does x_1+x_2+x_3+x_4≤ 5 have?
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