Exercise · ZW-COMBINATIONS-WITH-REPETITION-004 · ★★☆ Intermediate
How many positive integer solutions does \(x_1+x_2+x_3=7\) have?
Choose one option.
Show answer
- C. 15
Explanation
With \(y_i=x_i-1\ge 0\) one gets \(y_1+y_2+y_3=4\), which has \(\binom{4+3-1}{4}=\binom{6}{4}=15\) solutions. The 36 counts the nonnegative ones.
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-003 How many nonnegative 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?
- ZW-COMBINATIONS-WITH-REPETITION-008 In how many ways can 10 identical marbles be distributed into 4 distinct boxes, allowing empty boxe…
- 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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