Question

How many ways are there to change a $10 note into any mixture of $1 and $2 coins?

Answer 1

Since 10 dollars is a small sum, you can just count manually, i guess.
now i am gonna write down all the ways you can mix the $1 and $2 to get $10

2+2+2+2+2 --- (5 2's)
2+2+2+2+1+1 ---(4 2's and 1 1)
2+2+2+1+1+1+1 ---(3 2's and 4 1's)
2+2+1+1+1+1+1+1 ---(2 2's and 6 1's)
2+1+1+1+1+1+1+1+1 ---(1 2 and 8 1's)
1+1+1+1+1+1+1+1+1+1 --- (10 1's)

Essentially there are 6 ways.