Look up Rule of Product, or what I learned as the “multiplication counting principle”. It’s the root of all permutation and combination rules. It says that if there are a ways to make the first choice and b ways to make the second choice…z ways to make the 26th choice, the possibilities total to ab…*z.
I’ll try to make this intuitive. Imagine if you tried to write out all the combinations. Suppose I said to you “There are just seven hat colors. Write all the combinations out.” You’d have no problem with that. You’d just write:
R
O
Y
G
B
I
V
You’d see that you need 7 bears. But then I come up to you and say “Hey, we’ve just decided we have our bears to have vests. There are still seven colors, but they can’t be the same as the hat the bears already picked.” Well, you’d look at your single red bear and say “I’ll need more bears, so that I can put six vests on red-hatted bears.” So then you’d have:
RO, RY, RG, RB, RI, RV
O
Y
G
B
I
V
Hey, look. The single red bear became six unique bears. It’s almost like your stuffed animals are multiplying (see what I did there?) And then you’d guess that you’d do the same thing with the other bears. In fact, each single bear would become six bears.
Hmm…7 bears, each individual one ‘multiplies’ and becomes 6…that sounds an awful lot like 7*6 = 42.
So now you’ve got 42 unique bears sitting on your dressing table and I come up and say “Y’know, we like the hats, we like the vests, but now we want ribbons. Same colors, same rules as the vests. Kthxbai.” Now you’d have to figure out how many bears you’ll need.
So you take a single bear from the pile like the RO bear.
ROY, ROG, ROB, ROI, ROV
RY
RG
RB
RI
RV
...(and 36 others)
That single RO bear became five bears. It’s like it had babies…like it multiplied. By five. And you started with 42 bears. So each bear will become five bears. 42*5 = 210.
And that’s why the multiplication counting principle works. With every choice, every unique thing you already had multiplies by the amount of new choices there are.