[閒聊] 您累了嗎 看份考古題吧

[asdofe (沒有一天到晚在過年的啦)]

這是我讀ds讀到很煩很煩的時候無聊看ntu的考古題版的時候看到的

重點是第十一題...XDDD

真是有梗的老師XDDD

十分沒來上課就掰嚕~~~XD

像我就會掰= ="

 作者  oscarchichun (ㄍ一)                                     看板  NTU-Exam 
 標題  [試題] 98上 陳健輝 離散數學 第一次期中考                               
 時間  Wed Oct 28 21:14:03 2009                                               
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課程名稱︰離散數學
課程性質︰系必修
課程教師︰陳健輝
開課學院：電資學院
開課系所︰資訊系
考試日期（年月日）︰2009/10/28
考試時限（分鐘）：120
是否需發放獎勵金：是
（如未明確表示，則不予發放）

試題 :

範圍:Combinatorics

(For each problem, please provide your computation details,
 not only your answer.)

1. How many 8-digit quaternary (0, 1, 2, 3) sequences are there that
   they each contain exactly two 3's or none at all?            (10%)

2. Solve a[n+2] - 6a[n+1] + 9a[n] = 3*2^n + 7*3^n for n>=0

3. Compute the number of integer solutions for x1+x2+x3+x4 = 24,
   where 4<=xi<=9 for i = 1, 2, 3, 4.                           (10%)

4. In how many ways can 24 bottles of soft drinks, 12 of type A and 12
   of type B, be distributed among five surveyors so that each surveyor
   gets at least one bottle of type A and at least two bottles of
   type B?                                                      (10%)

5. Compute the number of ways to park motorcycles and compact cars in
   a row of n spaces, provided each cycle requires one space, each
   compact needs two, and all n spaces are used up. Assume that all
   cycles are identical in appearance, as are the cars.         (10%)

6. In how many ways can one arrange all of the letters INFORMATION so
   that no pair of consecutive letters (i.e., "IN", "NI", "IO", "OI",
   "NO", and "ON") occurs twice? For example, IINNOOFRMTA and FORTMAIINON
   are counted, but INFORINMOTA and NORTFNOIAMI are not counted, because
   the latter two contain "IN" and "NO", respectively, twice. Notice that
   it is impossible that two "IN"s and two "NI"s (similarly, for "IN" and
   "IO", "IN" and "ON", "NI" and "OI", "NI" and "NO", "IO" and "OI", "IO"
   and "NO", "OI" and "ON", and "NO" and "ON") occur simultaneously.
   You are perprmitted to write your answer as a conbination of some
   factorials, e.g., 11!+7*10!+5*8!=2*6!.                       (10%)


7. Two dice, denoted by R and G, are rolled six times. Under the condition
   of (R, G)不屬於{(1,2), (1,3), (2,3), (2,4), (4,5), (5,5)}, compute the
   number of ways for each of R and G to occur 1,2,...,6(for example, (1,2),
   (3,3), (2,6), (5,5), (4,1), (6,4) and (5,4), (1,6), (6,3), (3,5), (4,2),
   (2,1) are two ways).                                         (10%)

8. Define φ(n) to be the number of integers m so that 1<=m<=n and
   gcd(m, n) = 1, where n>=2 is an integer. For example, φ(2) = 1,
   φ(3) = 2, φ(4) = 2, and φ(5) = 4.
   Prove IN DETAIL that when n = (p1)^a *(p2)^b *(p3)^c,
   φ(n) = n(1-1/p1)(1-1/p2)(1-1/p3), where p1, p2, and p3 are three distinct
   prime numbers.                                               (10%)

9. Pauline takes out a loan of 200,000 dollars that is to be paid back in
   20 years with the interest rate 5% per year. The constant payment at
   the end of each year is (200,000/w)*([x*(1.05)^20]/[y*(1.05)^20 + z]).
   Please find w, x, y, and z.                                  (10%)

10. Solve b[n+1] = b[n] + b[n-1] +...+b[1] + b[0] with n>=0 and b[0] = 1.
                                                                (10%)

11. Please remember what (funny or impressive or encouraging or ...,
   but NOT RELATED TO DISCRETE MATHEMATICS) things did the professor Chen
   mention in the class. Write down any two of them.            (10%)


--
心中有變數，就不用變數代換法

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