標題:
permutation and combination
發問:
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1) A school organized a table-tennis tourament. Any two players must play against each other. However, a player was quit after playing 3 matches and another player was absent from two matches, therefore these two players did not play against each other. If a total of 57 matches were played finally, how many players... 顯示更多 1) A school organized a table-tennis tourament. Any two players must play against each other. However, a player was quit after playing 3 matches and another player was absent from two matches, therefore these two players did not play against each other. If a total of 57 matches were played finally, how many players were there orginally. 2) Evaluate nP2 (n+2)P4 - [(n+1)P3]^2 3) Find the number of sides of a convex polygon with 252 diagonals. thanks
最佳解答:
1) Let n be the number of players there were originally. (n - 1) players play all the matches (n-1C2). A player was quit after playing 3 matches (+3). Among the (n - 1) players, one of them was absent from 2 matches but 1 of the 2matches was against the player who was quit (-1). Number of matches: n-1C2 - 1 + 3 = 57 (n - 1)! / [(n - 1 - 2)! 2!] = 55 (n - 1)(n - 2) / 2 = 55 (n - 1)(n - 2) = 110 n2 - 3n + 2 = 110 n2 - 3n - 108 = 0 (n - 12)(n + 9) = 0 n = 12 Hence, the number of players there were originally = 12 ===== 2) nP2 x n+2P4 - (n+1P3)2 = [n! / (n - 2)!] x [(n + 2)! / (n + 2 - 4)!] - [(n + 1)! / (n + 1 - 3)!]2 = n(n - 1) x (n + 2)(n + 1)n(n - 1) - [(n + 1)n(n - 1)]2 = (n + 2)(n + 1)n2(n - 1)2 - (n + 1)2n2(n - 1)2 = (n + 1)n2(n - 1)2[(n + 2) - (n + 1)] = (n + 1)n2(n - 1)2 ===== 3. Let n be the number of sides of the convex polygon. No. of diagonals of the convex polygon : nC2 - n = 252 [n! / (n - 2)!2!] - n = 252 n(n - 1) / 2 = 252 + n n2 - n = 504 + 2n n2 - 3n - 504 = 0 (n - 24(n + 21) = 0 n = 24 or n = -21 (rejected) Hence, the number of sides of the convex polygon = 24
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