1.(15%) Find the number of permutations of a,b,c,...,x,y,z, in which none of the patterns 'spin','game','path',or 'net' occurs. 1.(15%) Find the number of permutations of a, b, c, ... , x, y,z, in which none of the patterns 'spin' , 'game' , 'path',or 'net' occurs 2.(15%) For the positive integers 1, 2, 3,..., n-1, n,there are 11660 derangements where 1, 2, 3, 4, 5, appear in the first five positions. what is the value of n? 3.(10%) Determine the coefficient of x^3 in 1/[(x-2)(x-3)]? 4.(10%) Find the exponential generating function for the sequence 0!, 1!, 2!, 3! .... 5.(10%) A ship carries 48 flags, 12 each of the colors red, white, blue and black.Twelve of these flag are placed on a vertical pole in order to communicate a signal to other ship. How many of these signal use an even number of blue and odd number of black flags? 6.(10%) Find a recurrence relation for the number of binary sequences of length n that have no consecutive 1's. 7.(15%) Solve the recurrence relation a(n+2) - 6a(n+1) + 9a(n) = 3*2^n + 7*3^n where n ≧ 0 and a(0) = 1 ,a(1) = 4. ***藍底白字括號部份代表下標字*** 8.(15%) The following table make (R,+,．) into a ring, where R ={s,t,v,w,x,y}. (1)What is the Zero for this ring? (2)What is the addition inverse of x? (3)Does R have a unity? (4)Is R a integral domain? (5)Is R a field? ┌─┬─────────┐ ┌─┬─────────┐ │＋│s t v w x y │ │．│s t v w x y │ ├─┼─────────┤ ├─┼─────────┤ │ s│s t v w x y │ │ s│s s s s s s │ │ t│t v w x y s │ │ t│s t v w x y │ │ v│v w x y s t │ │ v│s v x s v x │ │ w│w x y s t v │ │ w│s w s w s w │ │ x│x y s t v w │ │ x│s x v s x v │ │ y│y s t v w x │ │ y│s y x w v t │ └─┴─────────┘ └─┴─────────┘ -- 每次po完一篇題目 就會發現自己又寫錯了幾題...唉唉唉... --

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