Discrete Mathematics with Graph Theory (3rd Edition) 204

Discrete Mathematics with Graph Theory (3rd Edition) 204 -...

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202 Solutions to Exercises The Inclusion-Exclusion Principle (6.1.2) says I U~=l Ai I = Ei IAil- Ei<j IAi n Ajl + Ei<j<k IAi n Aj n Akl - ... + (-1)n+lIAl n A2 n··· n Ani Now IAll = (n - I)! (1 goes into the first position, the remaining numbers go anywhere in the remaining n - 1 positions) and, similarly, each Ai has cardinality (n - I)!. The set Al n A2 contains those permutations of 1, 2, ... ,n in which 1 and 2 are in the right positions; there are (n - 2)! of these corresponding to the arbitrary placement of 3,4, ... , n in the last n - 2 positions of a permutation. Similarly, each of the (~) terms IA n Aj I is (n - 2)!' Similar reasoning shows that each of the (~) terms IAi n Aj n Akl equals (n - 3)!, and so on. We get I iQ Ai 1= n(n -I)! - (;) (n - 2)! + (;) (n - 3)! - ... + (_I)n+1 (~)O! I n! n! ()n+l = n. - 2! + 3! - ...
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Unformatted text preview: + -1 and so D I [I n! n! ()n+l] ( 1 1 1 ()n 1 ) n = n. -n. -2! + 3! -... + -1 = n! 1- I! + 2! -3! + . .. + -1 n! as required. 10. (a) [BB] Using Proposition 7.6.2, Dn = n! -n! + [n(n -1)(n - 2) . . 3] -[n(n -1)(n - 2) ... 4] + . .. + (-It-l n + (_I)n = n[ (n - 1) (n - 2) ... 3 -(n - 1) (n - 2) ... 4 + . .. + (-1) n-l (n - 1) 1 + ( -1) n == (_I)n (mod n). (b) [BB] This follows immediately from part (a). 11. (a) [BB] (n -1)(D n - l + D n - 2 ) = (n -1) { (n -I)! [1 -~ + ~ - ... + (-1 t-l 1 ] I! 2! (n -I)! +(n-2)![I-~+~-&quot;'+(-I)n-2 1 ]} I! 2! (n -2)! =n(n-l)![I-~+~-&quot;'+(-lt-l 1 ] I! 2! (n -I)! -(n -I)! [1 -~ + ~ - ... + (-1 t-l 1 ] I! 2! (n -I)! +(n-l)![I-~+~-&quot;'+(-lt-2 1 ] I! 2! (n -2)! = n!(I-~ + ~ - ... + (-lt~) -n!(-lt~ -(n -1)!(-lt-l 1 I! 2! n! n! (n - I)! = Dn -(-I)n -(_I)n-l = Dn...
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