Counting l In fact, how common are nondefective dollars? Assuming that the digit portions of serial numbers all occur equally often, we could answer this question by computing fraction dollars that are nondefective of serial #'s with all digits different total# of serial#’s Let's first consider the denominator. Here there are no restrictions; there areare 10 possi ble first digits, 10 possible second digits, 10 third digits, and so on. Thus, the total number of 8-digit serial numbers is 10 by the Generalized Product Rule. (Alternatively, you could conclude this using the ordinary Product Rule; however, the Generalized Product Rule is strictly more powerful. So you might as well forget the orignial Product Rule now and free up some brain space for 6.002. Next, let's turn to the numerator. now we re not permitted to use any digit twice. S there are still 10 possible first digits, but only 9 possible second digits, 8 possible third digits, and so forth. Thus there are 10! 10.9.8·76·5·4·3 serial numbers with all digits different. Plugging these results into the equation above, we find 1,814,400 fraction dollars that are nondefective 100,000,000 1.8144% 1.2 A Chess Problem In how many different ways can we place a pawn(), a knight(k), and a bishop(b)on a chessboard so that no two pieces share a row or a column? a valid configuration is shown below on the the left, and an invalid configuration is shown on the right kCounting II 3 In fact, how common are nondefective dollars? Assuming that the digit portions of serial numbers all occur equally often, we could answer this question by computing: # of serial #’s with all digits different fraction dollars that are nondefective = total # of serial #’s Let’s first consider the denominator. Here there are no restrictions; there are are 10 possi￾ble first digits, 10 possible second digits, 10 third digits, and so on. Thus, the total number of 8­digit serial numbers is 108 by the Generalized Product Rule. (Alternatively, you could conclude this using the ordinary Product Rule; however, the Generalized Product Rule is strictly more powerful. So you might as well forget the orignial Product Rule now and free up some brain space for 6.002.) Next, let’s turn to the numerator. Now we’re not permitted to use any digit twice. So there are still 10 possible first digits, but only 9 possible second digits, 8 possible third digits, and so forth. Thus there are 10! 10 · 9 8 7 6 5 4 3 = · · · · · · 2 = 1, 814, 400 serial numbers with all digits different. Plugging these results into the equation above, we find: 1, 814, 400 fraction dollars that are nondefective = 100, 000, 000 = 1.8144% 1.2 A Chess Problem In how many different ways can we place a pawn (p), a knight (k), and a bishop (b) on a chessboard so that no two pieces share a row or a column? A valid configuration is shown below on the the left, and an invalid configuration is shown on the right. k b p p b k valid invalid