Surjections onto $3$ elements

#surjections#inclusion-exclusion#Stirling

Statement

Count surjections from a $5$ -element set to a $3$ -element set.

Hint

Use $3^5-\binom{3}{1}2^5+\binom{3}{2}1^5-\binom{3}{3}0^5$ .

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Inclusion-exclusion. $243-3\cdot32+3\cdot1-0=243-96+3=150$ .

Alternative. $3!\,S(5,3)=6\cdot25=150$ .

Conclusion. There are $150$ surjections.