Why does 6C2 = 6C4?
Here is Allison's explanation of the specific problem I was asking about.
And here is Anonymous explaining why 6C2 = 6C4:
Image you have six balls (red, orange, yellow, green, blue, and purple, for example). You also have a box.
6C2: I have six balls, and I am choosing which two balls to put in the box.
6C4: I have six balls, and I am choosing which four to leave out of the box.
Thus, the combination "I put red and orange in the box," is the same way of splitting up the balls as "I leave yellow, green, blue, and purple out of the box." Counting the ways to put two in the box is the same as counting the ways to leave four out of the box.