Ok to post! Thanks for the extension Dr. Z

Question 1

(i) The number of 10-letter words in the alphabet {1,4,6} that add-up to 201 
There are no 10 letter words in the alphabet that add up to 201

(ii) The number of words in the alphabet {2,3,7} of ANY length, that add-up to 411
-

(iii) The number of sequenes of any length whose entries are either 1 (mod 5) or 4 (mod 5) and that add up to 341

Question 2:
Using the convention of representing partitions as weakly-decreasing sequences, write a procedure, call it Pnk(n,k) that inputs non-negative integers n and k and outputs the SET of all partitions of n whose LARGET part is k.

Pnk:=proc(n,k) local ans, W, w, i:
option remember:
if n < 0 or k < 0 then 
 RETURN(FAIL):
fi:
if n = 0 and k = 0 then
 return {[]}:
fi:
if k > n then
 return {}:
fi:
ans := {}:
W := {}:
for i from 0 to k do
	W := W union Pnk(n-k,i):
end:
ans := {seq([k,op(w)], w in W)}:
end:

Question 3:
Count the numbers:
Pnk:=proc(n,k) local ans:
option remember:
if n < 0 or k < 0 then 
 RETURN(FAIL):
fi:
if n = 0 and k = 0 then
 return 1:
fi:
if k > n then
 return 0:
fi:
ans := {}:
W := {}:
for i from 0 to k do
	ans := ans+ Pnk(n-k,i):
end:
ans:
end:


Question 4:

pn:=proc(n) local ans, i:
if n <= 0 then
 RETURN(FAIL):
fi:
ans:= 0:
while i < n do
ans := ans+pnk(n,i):
i:=i+1
end:
ans

end:

Sequence generated: 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604
OEIS Number A41

Question 5:
[seq(pn(5*n + 4) mod 5, n = 1 .. 50)];
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

  0, 0, 0, 0, 0, 0, 0, 0]

[seq(pn(7*n + 5) mod 7, n = 1 .. 50)];
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

  0, 0, 0, 0, 0, 0, 0, 0]
  
[seq(pn(11*n + 6) mod 11, n = 1 .. 50)]
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]