Simple Combinations with Repetition Question 
In how many ways can we select five coins from a collection of 10 consisting of one penny, one nickel, one dime, one quarter, one half-dollar and 5 (IDENTICAL) Dollars ?

For my answer, I used the logic, how many dollars are there in the 5 we choose?
I added the case for 5 dollars, 4 dollars, 3 dollars and 2 dollars and 1 dollars and 0 dollars.
$$C(5,5) + C(5,4) + C(5,3) + C(5,2) + C(5,1) + 1 = 32$$
which is the right answer ... but there has to be a shorter way, simpler way. I tried using the repetition formula that didn't pan out.
If you could introduce me to a shorter way with explanation I appreciate it. 

Decide for each small coin whether you select that or not. Then top up with dollars until you have selected 5 coins in total.
The topping-up step does not involve any choice, so you have 5 choices to make, each with 2 options, giving $2^5=32$ combinations in all.
