0013409石婉甄
1.3-20
a.select 3 vertices from A,B,C,D,E,F,G,H in C(8,3) ways,so there are 56 ways
b.select 4vertices from A,B,C,D,E,F,G,H in C(8,4 ways,so there are70ays
c.total polygons 28-C(8,0)-C(8,1)-C(8,2)=219
1.3-25
b.4!0!2!=12
1.4-12
a.x1+x2+x3+x4+x5+x6=39,xi≧0,1≦i≦6.there are C(6+39-1,39)=C(44,39)
b.let yi=xi+3 1≦i≦5 y1+y2+y3+y4+y5≦54,yi≧0,C(6+54-1,54)=C(59,54)

1.4-19 r=4,nested for loop,so1≦m≦k≦j≦i≦20,we are making selections,with repetition of size r=4 from a collection of size n=20,=>C(23,4)
1.4-23
a.put one object into each container,then there are m-n identical object to place into distinct containersC(n+(m-n)-1,m-n)=C(m-1,m-n)=C(m-1,n-1)
b. put r object into each container. The remaining m-rn object canbe distributed among n distinct containers C(n+(m-rn)-1,m-rn)=(m-1+(1-r)n,n-1)
1.4-28
a.x1 1s followed by x2 0s followed by x3 1s followed by x4 0s followed by x5 1s followed by x6 0s,where x1+x2+x3+x4+x5+x6=n, x1,x6≧0 x2,x3,x4,x5>0
=>y1+y2+y3+y4+y5+y6=n-4,where yi≧0 for1≦i≦6
The number is C(6+(n-4)-1,n-4)=C(n+1,n-4)=C(n+1,5)
b. for n≧6,x1 1s followed by x2 0s followed by x3 1s followed by x4 0s followed by x5 1s followed by x6 0s…followed by x8 0s,where x1+x2+x3+x4+x5+x6+x7+x8=n, x1,x8≧0,x2,x3..x7>0
=>y1+y2+y3+y4+y5+y6+y7+y8=n-6,where yi≧0,for 1≦i≦8
The number is C(8+(n-6)-1,n-6)=C(n+1,n-6)+C(n+1,7)
c.2nstring in total n+1 strings where there are k 1s followed by n-k 0s for k=0,1,2…3.there are no o1s in n+1 strings,
2^n-(n+1)= 2^n-C(n+1,1)=>at least one 01
C(n+3,1) strings contain one occurrence of 01. C(n+5,1)strings with two occurrence, C(n+7,1)strings with three occurences, i. x1+x2+x3..+xn+1=n,x1,xn+1≧0,x2,x3,x4..xn>0
=>y1+y2+y3+..yn+1=n-(n-1)=1,where y1,y2..yn+1≧0
=>C((n-1)+1-1,1)=C(n+1,1)=C(n+1,n)=C(n+1,2(n-12) +1) ii. x1+x2+x3..+xn+2=n,x1,xn+2≧0, ,x2,x4..xn+1>0
=>y1+y2+y3+..yn+2=n-n=0,where yi≧0, 1≦i≦n+2
=>C((n+2)+0-1,0)=C(n+1,0)=C(n+1,n+1)=C(n+1,2(n/2) +1)

So 2n-C(n+1,1)=C(n+1,3)+C(n+1,5)+..+ C(n+1,n)/ C(n+1,n+1){n odd/n even}