Explanation Change the problem into a question of chess board with forbidden positions; X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 Table 1 The r k is the number of ways to place k non attacking rooks on the n × n board where each of the k rooks is in a forbidden position, ( k = 1 , 2 , .. , n ) . Consider the number of ways to place six non attacking rooks on the chessboard such that no rook is in a forbidden position. By Theorem 6.4.1, this number is ∑ k = 0 6 r k ( − 1 ) k ( 6 − k ) ! . Also know that r 0 = 1 and r 1 is the number of ways of placing 1 non-attacking rooks on the forbidden positions so, r 1 = 8 . To compute r 2 , two rooks have to be in two different rows. To compute r 3 three rooks have to be in three different rows etc. Now select pairs in which X are present. So, total number of pairs row are possible ( 1 , 3 ) , ( 1 , 4 ) , ( 1 , 5 ) , ( 1 , 6 ) , ( 3 , 4 ) , ( 3 , 6 ) , ( 4 , 6 ) . In whose two rooks are distribute in different rows with not repetition. The placement of two rooks 1 and 3 in pair form in 5 different ways. Suppose that if place first rook in X 1 box then second rooks place in X 3 box.(i.e. for first way to place of rooks). Same as other possible different ways place to rooks. And always mind that two boxes of X never come to below and up each other. Same as in other pairs the two rooks are distribute shown in the below given table

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Chapter 6, Problem 25E

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The count of permutations, i1i2i3i4i5i6 of {1,2,3,4,5,6} that are following the given condition.

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The count of permutations, i 1 i 2 i 3 i 4 i 5 i 6 of { 1 , 2 , 3 , 4 , 5 , 6 } that are following the given condition.

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The count of permutations, i1i2i3i4i5i6 of {1,2,3,4,5,6} that are following the given condition.

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Chapter 6 Solutions