Solutions from a student (dragged) 25

# Solutions from a student (dragged) 25 - N(2 = N(0 3 N(1 3...

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First Place Second Place Third Place A , B , C A , B C A , C B C , B A A B , C B C , A C A , B A B C B C A C A B A C B . . . . . . . . . B A C C B A Table 2: Here we have enumerated many of the possible ties tha tcanhappenw i thth ree people. The ±rst row corresponds to all three in ±rst place. The next three rows corresponds to two people in ±rst place and the other in second place. The third row corresponds to two people in second place and one in ±rst. The remaining rows correspond to one person in each position. The ellipses ( . . .) denotes thirteen possible outcomes. Part (d): For the speci±c case of N (3) we ±nd that N (3) = 2 ± j =0 ² 3 j ³ N ( j ) = ² 3 0 ³ N (0) + ² 3 1 ³ N (1) +
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Unformatted text preview: N (2) = N (0) + 3 N (1) + 3 N (2) = 1 + 3(1) + 3(3) = 13 . We also ±nd for N (4) that N (4) = 3 ± j =0 ² 4 j ³ N ( j ) = ² 4 ³ N (0) + ² 4 1 ³ N (1) + ² 4 2 ³ N (2) + ² 4 3 ³ N (3) = N (0) + 4 N (1) + 3 · 4 2 N (2) + 4 N (3) = 1 + 4(1) + 6(3) + 4(13) = 75 . Problem 17 (why the binomial equals the multinomial) The expression ² n r ³ is the number of ways to choose r objects from n , leaving another group of n-r objects. The expression ² n r, n-r ³ is the number of divisions of n distinct...
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