I have 12 balls labeled from 1 to 12 in a bag. I randomly draw 12 times from this bag. Every time I draw a ball, I append it to the right of a line. The balls in the line are ordered from the left, so the leftmost ball is first in line. At the end of this, a ball is in its proper position if its order matches its label. For instance, ball 5 is in its proper position if it's 5th in line.
What's the probability that 7 balls are in their proper positions?
I tried just taking ${12}\choose{7}$ since it does matter which specific orders have balls matched. However, I just realized that the draws depend on each other. For instance, if 11 balls are in proper positions, the last one must be in proper positions. What in probability can I latch onto to solve this problem?