How it works
Write the n identical objects as stars and use k − 1 bars to split them into k groups. Every arrangement of stars and bars is one distribution, so the number of ways to give n identical objects to k people (some may get none) is
C(n + k − 1, k − 1)
This is also the number of nonnegative integer solutions to x₁ + … + xk = n. If every person must get at least one, hand out one each first, then distribute the remaining n − k: that gives C(n − 1, k − 1). Stars and bars only works when the objects are identical; distinct objects need a different count.
Stars And Bars practice problems with solutions
These problems come from Mathyessy's question bank. Try each one before opening the solution.
Seven identical candies are given to 3 kids. How many distributions are possible?
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Answer: 36
7 stars and 2 bars: C(9, 2) = 9 × 8 ÷ 2 = 36.
Five identical candies are given to 2 kids. How many distributions are possible?
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Answer: 6
C(5 + 1, 1) = 6. You can also list them: the first kid gets 0, 1, 2, 3, 4 or 5.
How many positive integer solutions does x + y + z = 6 have?
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Answer: 10
Each variable is at least 1, so give 1 to each first. That leaves 3 to share among 3 variables with zeros allowed: C(5, 2) = 10.
How many nonnegative integer solutions does x + y + z + w + v = 5 have?
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Answer: 126
5 stars and 4 bars: C(9, 4) = 9 × 8 × 7 × 6 ÷ 24 = 126.
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