1. Since the balls are indistinguishable, we just need to keep track of how many balls in each box.
The possible arrangements are:
\((4, 0, 0); (3, 1, 0); (2, 2, 0); (2, 1, 1) \Rightarrow \boxed{4}\)
2. We start by splitting the numbers into three cases:
The one digit numbers don't have any zeroes.
The two digit numbers use two zeroes: 10 and 20.
There are \(3^2 = 9\) three-digit numbers starting with 1, and 9 starting with 2. For each leading digit, a zero appears in each digit in \(9\div3 = 3\) of the numbers, so each has a total of \(3 + 3 = 6\) zeroes. Thus, the 3-digit numbers contain \(2\cdot6 = 12 \) zeroes.
3. Here is the 3 by 3 grid:
R | B | G |
G | R | B |
B | G | R |
For the first column, we have 3! = 6 arrangements of red, green, and blue.
For the second column, we have only \(\binom21\) arrangements of red, green, and blue.
For the last column, we have only 1 arrangements of red, green, and blue.
For a total of 6 * 2 = 12 arrangements.
I hope this helped,
Gavin.