I'm also a visual learner, so here is what I think should help you visualize it.
These are the three spaces --> _ _ _
_ _ _
R R R
O O O
Y Y Y
G G G
B B B
I I I
P P P
These are the seven choices for each space
So lets cut it down to 2 spaces to make it easier
Each choice of the first space has 7 choices for the next space
the same for each number going down the list
R has 7 choices for the next space, O has 7 choices for the next space, and so on
=> This makes 7 + 7 + 7 + 7 + 7 + 7 + 7 = 7 * 7 = 7^2
So if you repeat this method for 3 spaces, you get 7^3 which equal 343 ways.
Hope this helps! :)
