ing ll an invalid configuration is shown on the right valid invalid Let a be the set of all sequences where ri and r2 are distinct rows and ci and c2 are distinct columns. let b be the set of all valid rook configurations. There is a natural function f from set A to set B; in particular, f maps the sequence(r1, C1, r2, C2)to a configuration with one rook in row r1, column Ci and the other rook in row r2, column c2 But now theres a snag. Consider the sequences (1,1,8,8) d(8,8,1,1) The first sequence maps to a configuration with a rook in the lower-left corner and a rook in the upper-right corner. The second sequence maps to a configuration with a rook in the upper-right corner and a rook in the lower-left corner. The problem is that those are two different ways of describing the same configuration In fact, this arrangement is shown on the left side in the diagram above More generally, the function f map exactly two sequences to every board configuration that is f is a 2-to-l function. Thus, by the quotient rule, A=2. B Rearranging terms 8·7)2 On the second line, weve computed the size of A using the General Product rule just as in the earlier chess problem 2.2 Knights of the Round Table In how many ways can King Arthur seat n different knights at his round table? Two lent if xample, the following two arrangements are equivalent:6 Counting II an invalid configuration is shown on the right. r r r r valid invalid Let A be the set of all sequences (r1, c1, r2, c2) where r1 and r2 are distinct rows and c1 and c2 are distinct columns. Let B be the set of all valid rook configurations. There is a natural function f from set A to set B; in particular, f maps the sequence (r1, c1, r2, c2) to a configuration with one rook in row r1, column c1 and the other rook in row r2, column c2. But now there’s a snag. Consider the sequences: (1, 1, 8, 8) and (8, 8, 1, 1) The first sequence maps to a configuration with a rook in the lower­left corner and a rook in the upper­right corner. The second sequence maps to a configuration with a rook in the upper­right corner and a rook in the lower­left corner. The problem is that those are two different ways of describing the same configuration! In fact, this arrangement is shown on the left side in the diagram above. More generally, the function f map exactly two sequences to every board configuration; that is f is a 2­to­1 function. Thus, by the quotient rule, |A| | = 2 · |B . Rearranging terms gives: |B = |A| | 2 (8 · 7)2 = 2 On the second line, we’ve computed the size of A using the General Product Rule just as in the earlier chess problem. 2.2 Knights of the Round Table In how many ways can King Arthur seat n different knights at his round table? Two seatings are considered equivalent if one can be obtained from the other by rotation. For example, the following two arrangements are equivalent: