Exercise · ZW-COMBINATIONS-WITH-REPETITION-008 · ★★☆ Intermediate
In how many ways can 10 identical marbles be distributed into 4 distinct boxes, allowing empty boxes?
Choose one option.
Show answer
- C. 286
Explanation
These are nonnegative solutions of \(x_1+\dots+x_4=10\): \(\binom{10+4-1}{10}=\binom{13}{3}=286\). The 210 is \(\binom{10}{4}\), which does not allow repetition.
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-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?
- 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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