Example: Use generating functions to determine the C(k+ )r=2n(n=0,12…) r=2n+1(n=0,1 Solution: Let generating functions of ( b be f(y) f(y)=(ly+y2+.,ty)(1+yy2+…,tyn).(1+y+y2+,…,+yy) Example: Let s=oooa1,00a2., ooag. Determine the number of r-combinations of s so that each of the k types of objects occurs even times. Solution: Let generating functions of (b be f(y) f(y)=(1+y2+y4+.)k=1/(1y2)k =1+ky2+C(k+1,2)y4+.+C(k+n-1,n)y2n+▪ Example: Use generating functions to determine the number of r-combinations of multiset S={n1·a1 ,n2·a2 ,…,nk·ak }. ▪ Solution: Let generating functions of {br } be f(y)， ▪ f(y)=(1+y+y2+…+yn1)(1+y+y2+…+yn2)…(1+y+y2+…+ynk) ▪ Example: Let S={·a1 ,·a2 ,…,·ak }. Determine the number of r-combinations of S so that each of the k types of objects occurs even times. ▪ Solution: Let generating functions of {br } be f(y)， ▪ f(y)=(1+y2+y4+…)k=1/(1-y 2 ) k ▪ =1+ky2+C(k+1,2)y4+…+C(k+n-1,n)y2n+…    = + = + − = = = 0 2 1( 0,1, ) ( 1, ) 2 ( 0,1, ) 2 2   r n n C k r n n a r r r