26 Combinatorial Analysis Properties 26.17 The Twelvefold Way 26.19 Mathematical Applications

§26.18 Counting Techniques

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Keywords:: counting techniques
Notes:: See Riordan (1958, pp. 50–65).
Referenced by:: §26.17
Permalink:: http://dlmf.nist.gov/26.18
See also:: Annotations for Ch.26

Let $A_{1},A_{2},\ldots,A_{n}$ be subsets of a set $S$ that are not necessarily disjoint. Then the number of elements in the set $S\setminus(A_{1}\cup A_{2}\cup\cdots\cup A_{n})$ is

26.18.1		$\left\|S\setminus(A_{1}\cup A_{2}\cup\cdots\cup A_{n})\right\|=\left\|S\right\|+% \sum_{t=1}^{n}(-1)^{t}\sum_{1\leq j_{1}<j_{2}<\cdots<j_{t}\leq n}\left\|A_{j_{1% }}\cap A_{j_{2}}\cap\cdots\cap A_{j_{t}}\right\|.$
ⓘ Symbols: $\left\|\NVar{A}\right\|$ : number of elements of a finite set $A$ , $\cap$ : intersection, $\setminus$ : set subtraction, $\cup$ : union, $j$ : nonnegative integer, $n$ : nonnegative integer, $A_{k}$ : subsets and $S$ : set Permalink: http://dlmf.nist.gov/26.18.E1 Encodings: TeX, pMML, png See also: Annotations for §26.18 and Ch.26

Example 1

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See also:: Annotations for §26.18 and Ch.26

The number of positive integers $\leq N$ that are not divisible by any of the primes $p_{1},p_{2},\ldots,p_{n}$ (§27.2(i)) is

26.18.2		$N+\sum_{t=1}^{n}(-1)^{t}\sum_{1\leq j_{1}<j_{2}<\cdots<j_{t}\leq n}\left% \lfloor\frac{N}{p_{j_{1}}p_{j_{2}}\cdots p_{j_{t}}}\right\rfloor.$
ⓘ Symbols: $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $j$ : nonnegative integer, $n$ : nonnegative integer, $N$ : set and $p_{n}$ : primes Permalink: http://dlmf.nist.gov/26.18.E2 Encodings: TeX, pMML, png See also: Annotations for §26.18, §26.18 and Ch.26

Example 2

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See also:: Annotations for §26.18 and Ch.26

With the notation of §26.15, the number of placements of $n$ nonattacking rooks on an $n\times n$ chessboard that avoid the squares in a specified subset $B$ is

26.18.3		$n!+\sum_{t=1}^{n}(-1)^{t}r_{t}(B)(n-t)!.$
ⓘ Symbols: $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer, $r_{j}(B)$ : number of rook positions and $B$ : subset Permalink: http://dlmf.nist.gov/26.18.E3 Encodings: TeX, pMML, png See also: Annotations for §26.18, §26.18 and Ch.26

Example 3

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See also:: Annotations for §26.18 and Ch.26

The number of ways of placing $n$ labeled objects into $k$ labeled boxes so that at least one object is in each box is

26.18.4		$k^{n}+\sum_{t=1}^{n}(-1)^{t}\genfrac{(}{)}{0.0pt}{}{k}{t}(k-t)^{n}.$
ⓘ Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : nonnegative integer and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/26.18.E4 Encodings: TeX, pMML, png See also: Annotations for §26.18, §26.18 and Ch.26

Note that this is also one of the counting problems for which a formula is given in Table 26.17.1. Elements of $N$ are labeled, elements of $K$ are labeled, and $f$ is onto.

For further examples in the use of generating functions, see Stanley (1997, 1999) and Wilf (1994). See also Pólya et al. (1983).

26.17 The Twelvefold Way 26.19 Mathematical Applications

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§26.18 Counting Techniques

Example 1

Example 2

Example 3

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