#HW 11 Eshaan Gandhi. Ok to post
# Question 1 In how many ways can you assemble 11 Russian dolls of different sizes into 5 different "towers"? In how many ways can you do it with any number of towers.
#Answer 1: What we are looking for is the Sterling numbers of the second time. I find that using the maple procedure Snk and then to answer the second question, we've been asked the bell number and I use Bell3 procedure
Snk(11, 5)
                             246730
Bell3(11)
                             678570
#246730 ways to assemble 11 Russian dolls in 5 different towers and 678570 ways to do it in any number of towers
#Question 2 - First Write a one-line procedure xn(x,n) that inputs a variable x and a non-negative integer n and outputs
#x*(x-1)*(x-2)*...*(x-(n-1))
#using this, write another one-line procedure, call it Axn(x,n) that inputs a variable x and a non-negative integer n and outputs
#Sum(Snk(n,k)*xn(x,k),k=0..n)


xn:=proc(x, n) local ans, i, l1:
l1:=[seq(i, i=0..n-1)]:
ans := 1:
for i in l1 do:
ans:=ans*(x-i):
od:
ans
end proc:

Axn:=proc(x, n) local  i,k, l1, l2:
option remember:
l2:=0:
l1:=seq(Snk(n, k)*xn(x, k), k=0..n):
#l2:=seq(, k=0..n):
for i in l1 do:
l2:=l2+i:
od:
l2
#Snk(n,k)*xn(x,k),k=0.
end proc:

expand(Axn(x, 7))
                                7
                               x 
#Axn(x, n) = x^n

cnk:=proc(n,k)  option remember:
if n=1 then
 if k=1 then
 RETURN(1):
 else
 RETURN(0):
 fi:
fi:

cnk(n-1,k-1)+(n-1)*cnk(n-1,k):
end:

cnk(5,1)*x^1
                              24 x