Fw: [試題] 100下 呂育道 離散數學 第一次期中考
※ [本文轉錄自 NTU-Exam 看板 #1FT1-Q4J ]
作者: peter506g (一氧化二氫) 看板: NTU-Exam
標題: [試題] 100下 呂育道 離散數學 第一次期中考
時間: Thu Mar 29 16:38:47 2012
課程名稱︰離散數學
課程性質︰資訊系選修
課程教師︰呂育道
開課學院：電資學院
開課系所︰資訊系
考試日期（年月日）︰101/3/29
考試時限（分鐘）：180分鐘
是否需發放獎勵金：
（如未明確表示，則不予發放）
是
試題 :
12 13
Problem 1 (10 points) What is the coefficient of x y in the expansion
25
of (2x + 3y) ?
Problem 2 (10 points) Show that for all positive integers m and n,
m+n m+n
n ( ) = (m + 1)( )
m n+1
Problem 3 (10 points) Prove that if n is a nonnegative integer, then
2n n n 2
( ) = Σ ( )
n k=0 k
Problem 4 (20 points) Let A → B denote the set of functions from set A to
n
set B. (a) [10 points] How many functions in {0,1,2} → {0,1,2} are there?
n m
(b) [10 points] How many functions in ({0,1,2} → {0,1,2}) → {0,1,2} are
b b
a (a )
there? (Do not write somthing like x as it is ambiguous. Write x
b
a
or (x ) .)
Problem 5 (10 points) Prove the mutinominal theorem: If n is a positive
integer, then
n n1 n2 nm
(x1+x2+…+xm) = Σ C(n;n1,n2,…,nm)x1 x2 …xm
n1+n2+…+nm=n
where
n!
C(n;n1,n2,…,nm) = ───────
n1!n2!…nm!
+
Problem 6 (10 points) Prove that for any positive integer for n 屬於 Z
n Fi-1 Fn+2
Σ─── = 1 - ───
i=1 i n
2 2
Problem 7 (20 points) Consider
(p V q) → r
1) Give an equivalent statement without →
2) Is it a tautology?
3) Is it a contradiction?
4) Negate the result in (1) first and apply DeMorgan's laws to move the
negation connective to the primitive statements p,q,r.
Problem 8 (10 points) A function f : A → B is called one-to-one if
each element of B appears at most once as the image of an element in A.
Assume A = {a1,a2,…am}. How many one-to-one functions from A to B are
there if |A| = m and |B| = n ≧ m?
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