# Invention of Principle of Inclusion-Exclusion

I know the principle of inclusion-exclusion and solved some problems related to this principle but I don't know how this principle was discovered? Whenever I use this principle I was amazed thinking of how the concept of the principle of inclusion-exclusion came into the mathematicians head? Can anyone tell me how mathematician discovered the principle of inclusion-exclusion in combinatorics? What problems led them to introduce the the principle of inclusion-exclusion?

• Although this doesn't answer your question, I think the IEP is really quite natural. By drawing Venn diagrams, the case $n=2$ is obvious, the case $n=3$ is easy, and by then it's clear that there must be some generalization for all $n$ that can be found similarly. More relevant to your question is that in France the result is often called Poincaré's formula. I believe he proved it in his textbook on probability, though the result may have ben known earlier. I suspect it has been independently proven many times. – Jack M Feb 16 '15 at 13:15
• In his Enumerative Combinatorics, Richard Stanley quotes P. Stein as saying the inclusion-exclusion principle “is doubtless very old; its origin is probably untraceable.” – Jair Taylor Feb 24 '15 at 7:04

• The principle also made its way to recreational mathematics in the late 19th century: say that 70 per cent lost an eye- 75 per cent an ear- 80 per cent an arm- 85 per cent a leg- that'll do beautifully. Now, my dear, what percentage, at least, must have lost all four?" (Lewis Carroll, The Tangled tale, Knot X, p. 69, London, Macmillan and co., 1885) – Margaret Friedland Jan 8 '16 at 20:01