Exercise · ZW-COMBINATIONS-WITH-REPETITION-007 · ★★☆ Intermediate
How many distinct monomials of degree 3 are there in the variables \(x,y,z\)?
Choose one option.
Show answer
- D. 10
Explanation
Each monomial \(x^ay^bz^c\) with \(a+b+c=3\) is a selection of 3 variables with repetition: \(\binom{3+3-1}{3}=\binom{5}{3}=10\). The 27 counts with order.
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
- ZW-COMBINATIONS-WITH-REPETITION-001 An ice cream shop has 3 flavors. In how many ways can 5 scoops be ordered, repeating flavors if des…
- ZW-COMBINATIONS-WITH-REPETITION-002 3 donuts are chosen from 5 available types, with repetition and without order. How many selections…
- 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-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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