Inclusion Exclusion principle for coloring a board The questions says, how many ways are there to color a 3 by 3 board with colors red and blue, so that there are no 2 by 2 red squares.
Since I'm required to solve the problem using inclusion-exclusion principle only, I started by making a 3 x 3 board and analyzing how many 2 x 2 boards can it consist of, 
+---+---+---+  +---+---+---+
| 0 | 0 |   |  |   | 1 | 1 |
+---+---+---+  +---+---+---+
| 0 | 0 |   |  |   | 1 | 1 |
+---+---+---+  +---+---+---+
|   |   |   |  |   |   |   |
+---+---+---+  +---+---+---+

+---+---+---+  +---+---+---+
|   |   |   |  |   |   |   |
+---+---+---+  +---+---+---+
| 2 | 2 |   |  |   | 3 | 3 |
+---+---+---+  +---+---+---+
| 2 | 2 |   |  |   | 3 | 3 |
+---+---+---+  +---+---+---+

I assumed that only 4 of the 2 x 2 boxes can be possibly made.
There's 4 possible conditions which could lead to a 2 x 2 box with only red color.
So, I make the 4 conditions 
$C_1 = C_2 = C_3 = C_4$ = 2 x 2 box has red color 
N($\bar{C_1}\bar{C_2}\bar{C_3}\bar{C_4}) = S_0 - S_1 + S_2 - S_3 + S_4$
Where $S_0$ is all the possible ways to fill a 3 x 3 board with 2 colors.
$S_1=N(C_1) + N(C_2) + N(C_3) + N(C_4)$
$S_2=N(C_1C_2) + N(C_1C_3) + N(C_1C_4) + N(C_2C_3) + N(C_2C_4) + N(C_3C_4)$
and so on according to the notation of generalization of inclusion exclusion principle. 
$\bar{N} = N(\bar{C_1}\bar{C_2}...\bar{C_t}) = N - \sum_{(1≤i≤t)}N(C_i) + \sum_{(1≤i<j≤t)}N(C_iC_jC_k) + ...$
=> $S_0 = 2^9$
Because there's 4 possible choices and 2 colors to pick from, using the pigeon hole principle
=> $S_1$ = $4 \choose 1$ $4 + 2 - 1 \choose 2$   
=> $S_2$ = $4 \choose 2$$3 + 1 - 1 \choose 1$
=> $S_3 = S_4 = 0$
I wanted to know if I'm on the right path

To carry on from where you seem to have left the track a bit:
$S_1=N(C_1) + N(C_2) + N(C_3) + N(C_4)$
$S_2=N(C_1C_2) + N(C_1C_3) + N(C_1C_4) + N(C_2C_3) + N(C_2C_4) + N(C_3C_4)$
$S_3=N(C_1C_2C_3) + N(C_1C_2C_4) + N(C_1C_3C_4) + N(C_2C_3C_4) $
$S_4=N(C_1C_2C_3C_4) $
$N(C_i)$ involves colouring the requisite block red and then free choices for the other five squares, giving $N(C_i) = 2^5$ options and thus $S_1 = 4\cdot 2^5 = 128$
There are two varieties when looking at the components of $S_2$: adjacent and diagonal blocks. These give respectively $3$ and $2$ free squares and there are four adjacent and two diagonal cases, giving $S_2=4\cdot 2^3+2\cdot 2^2 = 32 + 8 = 40$
The components of $S_3$ only have one free square each, so $S_3=4\cdot 2 = 8$
Finally there is only one way to colour every square red as required to make all four blocks red.
Then as you observe, $S_0=2^9$ and the inclusion-exclusion calculation gives:
$R = 512-128+40-8+1 = 417$ cases where there is no red $2\times 2$ block

Cross-checking by a different method; we will have no red block if the centre square is blue, so that is $A_1=2^8=256$ possibilities immediately.
Then if the centre square is red, we can avoid red blocks if opposite mid-side squares are blue. There's a little inclusion-exclusion exercise to get that $A_2=2\cdot 2^6-2^4 = 112$
Then with one mid-side square blue but the opposite red, we can avoid red blocks either by having an adjacent mid-side and the remaining corner blue or having the opposite corners blue.  This works out to $A_3 = 4\cdot2^3+4\cdot2^2 = 48$
Finally if everything else is red and the four corners are blue the red blocks are also defeated, $A_4=1$.
Giving $R = 256+112+48+1 = 417$ again.
