Mathematical Experiment 7

exp7.mws

  1. List all subsets of {1,2,3,4,5}.
  2. List all permutations of [1,2,3,4,5].
  3. Express the number 10 into sums of positive integers without regards to order. For example:
  4. 10 = 2+7+1
    is the same as
    10 = 1+2+7.
  5. Express the number 10 into sums of positive integers with regards to order. For example:
  6. 10 = 2+7+1
    is different from
    10 = 1+2+7.
  7. How many subsets does {1,2,¼,70} have?
  8. How many permutations does [1,2,¼,70] have?
  9. How many ways to express the number 70 as a sum of positive integers without regards to order?
  10. How many ways to express the number 70 as a sum of positive integers with regards to order?
  11. List all possible values of
  12. x3-y2+xz
    where [ x,y,z] is a permutation of [4,5,6].
  13. Here is a systematical generation of all permutations of [1,2,¼,n].
 
visit:=proc(k) local t; global v,n; 
if k=n then print(v) else for t from  k to n 
do v:=subs({v[t]=v[k],v[k]=v[t]},v);
visit(k+1);
v:=subs({v[t]=v[k] ,v[k]=v[t] },v); od;
fi; 
end;
n:=5;
v:=[1,2,3,4,5];
visit(1);
Reference: G. Brassard and P. Bratley, Algorithmics, Theory and Practice, p.186.


exp7.tex