Zoom International Math League

When doing probability with objects, typically we need to consider different objects `distinguishable' even if the they look the same.

My favorite example of this: Suppose a bag has 1 red ball and 9 identical green balls. What is the probability you pick a green ball?

One could argue that there are only 2 outcomes here, pick a red ball or pick a green ball (since they're all identical). However, this does not mean the probability is 1/2 to pick a green ball, because there are more green balls. Note whether the green balls are identical or not doesn't matter here, and we need to consider them different / distinguishable for probability purposes to get the correct answer of 9/10.

The 2 Jacks in the card problem are similar, since they are different cards we consider them distinguishable when doing the counting, giving the 7! total ways that 7 cards can be dealt.