Chapter 6 The principle of inclusion and exclusion

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Chapter 6 The principle of inclusion and exclusion

• 6.1 ????
• (the principle of inclusion and exclusion)
• 6.2 ??(derangements)
• 6.3 ????? (the rook polynomial)

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6.1 ????
Determine the number of positive integers n , 1?n
?30, that are divisible by 2, 3, or 5.
Ans 15106-5-3-2122
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6.1 ????
?1. (a) Find the number of integers between 1 and
250 that are not divisible by any of the integers
2, 3, 5, and 7. (b) the number of integers that
not divisible by 2 nor by 7 but are divisible by 5
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6.1 ????
?2. Find the number of r-digit quaternary
sequences in which each of the three digits 1, 2,
and 3 appears at least once.
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6.2 ??(derangements)
A permutation of these integers is said to be a
derangement of the integers if no integer appears
in its natural position.
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6.2 ??(derangements)
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6.2 ??(derangements)
?1. Let n books be distributed to n children. The
books are returned and distributed to the
children again later on. In how many ways can the
books be distributed so that no child will get
the same book twice?
?2. In how many ways can the integers 1, 2, 3, 4,
5, 6, 7, 8, and 9 be permuted such that no odd
integer will be in its natural position?
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6.2 ??(derangements)
?3. Find the number of permutations of the
letters a, b, c, d, e, and f in which neither the
pattern ace nor the pattern fd appears.
6!-4!-5!3!
?4. In how many ways can the letters x, x, x, x,
y, y, y, z, and z be arranged so that all the
letters of the same kind are not in a single
block?
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6.3 ????? (the rook polynomial)
The problem of nontaking rooks a rook is a
chessboard piece which captures on both rows
and column
Given a chessboard, let rk denote the number of
ways of placing k nontaking rooks on the
board. r0 1 r1 4 r2 3 R(x) 1 4x3x2
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6.3 ????? (the rook polynomial)
Expansion formula
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6.3 ????? (the rook polynomial)
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6.3 ????? (the rook polynomial)
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6.3 ????? (the rook polynomial)
Given a chessboard C, find the rook polynomial of
C.