Splitting 20 thousand dollars among five funds

Question

You want to invest $20,000 across five mutual funds, putting money in only in multiples of $1,000. Counting two plans as different if any fund receives a different amount, how many allocation plans are there?

Short answer

$10{,}626$ . You are distributing $20$ identical units into $5$ funds, so stars and bars gives $\binom{24}{4}=10{,}626$ (allowing a fund to receive nothing). If every fund must receive at least $1,000 the count is $\binom{19}{4}=3{,}876$ .

Adapted from Crack, Heard on the Street, Q4.31, p.43; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Counting and Combinatorics
Sample Spaces, Events, and the Probability Axioms
Conditional Probability, Total Probability, and Bayes' Rule

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