Sums II
Sums and cases that can be expressed as a sum
1. Integers 0 to 9 are each written on a card. The 6 and 9 are different cards and not interchangeable. Cards 0, 1, 2, and 3 have different colors on either sides, so they each represent two different states, as opposed to each of the remaining 4 to 9 cards that only represents a single state.
How many different sequences are possible using 7 of these cards in a line?
Hint 1
Write a summation expressing all the cases.
Hint 2
The decisions are 1) how many double-sided cards to use, 2) how many ways to choose the hards, 3) which sides of the double-sided cards to use, and 4) how many ways to permute the cards.
Hint 3
Recall Method 1 from Permutations II: restrictions, #2.
2. Find the sum of all positive integers under 6000 that only uses the digits 0 to 5.
Hint 1
Consider all positive integers under 1000 as if they start with one or more 0s, such that all numbers to be added are "four digits".
Hint 2
Change the order to adding all the digits separately. Eg $5212 = 5000 + 200 + 10 + 2$.
Hint 3
216 of such integers have a 1 in the hundreds place.