1.
11 russian dolls into 5 towers:
Snk(11,5)
246730
11 russian dolls into any tower:
Bell3(11)
678570

2.
xn:=proc(x,n) local f,i:
f:=x:
for i from 1 to n-1 do
  f:=f*(x-i):
od:
f:
end:

Axn:=proc(x,n):
add(seq(Snk(n,k)*xn(x,k),k=0..n)):
end:

seq(Axn(x,i),i=1..20)
x, x^2, x^3, x^4, x^5, x^6, x^7, x^8, x^9, x^10, x^11, x^12, x^13, x^14, x^15, x^16, x^17, x^18, x^19, x^20

Axn(x,n)=x^n

3. 
Consider a permutation of n with k cycles. There are two cases to consider: first is when the element n belongs to its own cycle,
so removing it would leave c(n-1,k-1). The second case is n belongs to a cycle with other elements, so removing it would leave 
k cycles. There are n-1 ways we can insert n into it, so the second case is (n-1)*c(n-1,k). Combining two cases would give c(n,k).

4.
c:=proc(n,k)  option remember:
if n=1 then
 if k=1 then
 RETURN(1):
 else
 RETURN(0):
 fi:
fi:

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

5.
x*(x+1)*(x+2)*...*(x+n-1)