How many ways are there to place $7$ distinct balls into $3$ distinct boxes?
is the question I'm confused about.
The solution shows that the correct answer is $3^7$. I'm just confused why this is.
My thinking is that if there are 3 boxes, and 7 possible balls for each box:
number of choices: 7 6 5
individual boxes: _ _ _
So $7*6*5$ total possibilities...
But clearly, the logic in this problem is the following:
Number of choices: 3 3 3 3 3 3 3
Individual balls: _ _ _ _ _ _ _
Why is the 1st solution incorrect?