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  "body": "\n## Statement\n\nLet $A_1, \\dots, A_n$ be finite sets.\nFor a nonempty set $I \\subseteq \\{1, \\dots, n\\}$, let $A_I = \\bigcap_{i \\in I} A_i$.\nThen\n$$\\left| \\bigcup_{i=1}^{n} A_i \\right| = \\sum_{\\varnothing \\neq I \\subseteq \\{1, \\dots, n\\}} (-1)^{|I| + 1} |A_I|.$$\n\n## Remarks\n\nFor $n = 2$ this is [[inclusion–exclusion for two sets|0C4YEQ9H5E]].\nA common use counts the elements of a finite set $U$ that are in none of the sets $A_i \\subseteq U$. This number is $\\sum_{I} (-1)^{|I|} |A_I|$, over all $I$, with $A_\\varnothing = U$.\nThe [[formula for the number of derangements|03RSJXF728]] follows in this way.\n\n## Explanations\n\n### L1 human\n\nAdd the sizes of the single sets, subtract the sizes of the pairwise intersections, add the sizes of the triple intersections, and so on.\nEach element of the union is then counted exactly once in total.\n",
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