VID045_Probability, combinations, and permutations of chocolate candy in heart

In today's education video, I will show you 11 pieces of chocolate candy in a heart-shape container. There are seven different flavors. Then I will answer three math questions:
Question #1, how many combinations can be made with one serving (note that one serving = two pieces of chocolate)? To solve this question, we first need to know how many choices are there if you pick type A first. There are seven other choices you could make from seven different flavors. Next we need to find the number of choices if you pick type B first. (Note that you cannot pick type A again, since we just figured out that.) There are six more choices, so add six. Next if we pick type C first, then we have five other choices. If we pick type D first, then we have four other choices. Lastly, if the only pieces we are picking is unique, then there are three other choices (AB, AC, BC). 7 + 6 + 5 + 4 + 3 = 25, so there are 25 possible combinations.
Question #2, what are the chances of you picking a piece of chocolate that is unique on your first or second attempt? First of all, there is a 1/11 chance of you picking the correct piece on your first attempt, and a 10/11 chance that you will not. In the second attempt, since you took out one piece, there are 10 pieces left, and one of them is correct (1/10), so there is now a 9/10 chance you will not pick the correct piece. 10/11 * 9/10 = 90/110 = 9/11 (this is the probability that you will NOT pick the correct piece). So the chances that you will is 1 - 9/11, or 2/11 (about 18%). What are the chances of you picking a piece that is not unique on your first or second attempt? First of all, there is a 2/11 chance of you picking the correct piece on your first attempt, and a 9/11 chance that you will not. In the second attempt, there is only 10 pieces left, and there is now a 8/10 (or 4/5) chance that you will not pick the correct piece. 9/11 * 4/5 = 36/55 (this is the probability that you will NOT pick the correct piece). So the chances that you will is 1 - 36/55, or 19/55 (about 35%).
Question #3, how many different ways can the pieces of chocolate be arranged in the heart? This is the most tricky question, and it will take about a year for you to play all the over million of possible permutations. There is a much simpler way for you to figure out the number of permutations, with math. First, treat all of pieces of chocolate as different from each other, even if they are the same. There are 11 pieces that can fit in the first space, 10 for the second space, 9 for the third space, etc. 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 11! = 39,916,800. The ! symbol after the number means to take the factorial of. Now we can spot four pairs of chocolate that are the same, so this number will be halved four times. You can switch the same piece of chocolate, and there are no differences. So to half a number four times is the same as dividing it by sixteen (2! * 2! * 2! * 2! = 16). 39,916,800 / 16 = 2,494, 800 possible permutations.