combinatory

Question

let S be the set of natural numbers whose digits are chosen from {1,3,5,7} such that no digits are repeated find

1) |S|

2) $\sum (n)$ where n belongs to s

first part i know 64 numbers would be there how to approach 2nd part

Answer 1

length 1: 1 or 3 or 5 or 7 $\sum \left ( 1+3+5+7 \right )$=16

length 2: 13,15,17,31,35,37,51,53,57,71,73,75=$\sum \left (13,15,17,31,35,37,51,53,57,71,73,75 \right )$=527

similarly

length 3: 10656

length 4: 106656

Comments

we can go for this manual method if set given is small but what if the set given is large then we can not go for this method as we may skip some strings is there any other method here which ensures correct  solution in any case

---

currently i am not able to think of any other method, and to make sure that you don't miss any case use tree method.

like fix 1 then 3 branches are possible 3,5,7 now fix 3, then 2 branches are possible 5,7 fix 5 then 7,3 and fix 7 then 5,3 now repeat for other numbers