由 訪客 於 星期二 十月 28, 2008 12:55 am
不好意思上面的題目沒有打的很好,我在這裡再發一次
n+m r m n
1. Prove that C =ΣC C
r j=0 j r-j
2.
The following identity is known as Fermat’s combinatorial identity.
n n n-1
C =ΣC ,N≧K
k I=k k-1
Give a combinatorial argument (no computations are needed) to establish this identity.
Hint: Consider the set of numbers 1 through n. How many subsets of size k have I as their highest-numbered member?
3.
From a set of n people a committee of size j is to be chosen, and from this committee a subcommittee of size i, i<xml><v> <v></v><v><v></v><v></v><v></v><v></v><v></v><v></v><v></v><v></v><v></v><v></v><v></v><v></v></v><v></v><xml><o></o></v><v><v></v></v> j, is also to be chosen.
2.
Show that the probability that exactly one of the events E or F occurs equals P(E)+P(F)-2P(EF).
3.
If A<v> <v></v></v>B, express the following probabilities as simply as possible:
P(A|B), P(A|B’), P(B|A), P(B|A’)
4.
Let A,B,C be events relating to the experiment of rolling a pair of dice.
(1)
If P(A|C)>P(B|C) and P(A|C’)>P(B|C’) either prove that P(A)>P(B) or give a counterexample by defining events A,B,C for which it is not true.
(2)
If P(A|C)>P(A|C’) and P(B|C’)>P(B|C’) either prove that P(A)>P(B) or give a counterexample by defining events A,B,C for which it is not true.
Hint: Let C be the event that the sum of a pair of dice is 10; let A be the event that the first die lands on 6; let B be the event that the second die lands on 6.
Let S be a given set. If, for some k>0, S1,S2…Sk are mutually exclusive
k
YSi =S
i=1
, then we call the set {S1,S2,…,Sk} a partition of S. Let Tn denote the number of different partitions of {1,2,…,n}. Thus,T1=1(the only partition being S1={1}), and T2=2(the two partitions being {{1,2}},{{1},{2}}).
(1)
Show, by computing all partitions, that T3=5,T4=15.
(2)
Show that
n n
Tn+1=1+Σ C T
k=1 k k
以上,請教各位大大