It would be 12 x 11 because from those 12 people left (Kyra excluded because she is definitely is on the team), each of the 12 scenarios will branch into 11 branches because after choosing the person that will occupy the second seat (first seat is definitely Kyra's), we have to choose one person out of 11 people left to occupy the third seat. Let's first think about the total number of permutations with Kyra in it. If you calculate it mathematically (which we are), you can see that it's intuitive: So, let's think about why the number of teams with Kyra in it is 12C2.ġ:36 "If we know that Kyra is on a team, then the possibilities are who is going to be the other two people on the team." So, we pick from 12 people (Kyra is excluded because she is definitely will be on the team and we are calculating the PROBABILITY of the combinations of the other 2 people that will go with Kyra) to choose the other 2 people that will be with Kyra on the team. Basically we are counting the permutations of a single random group because all group will have the same number of permutations (that is because we have the same number of seat that they can take on in a single group). So, because there will be 3 people in a group, the number of arrangements in a single group is just 3! (3 factorial). ![]() ![]() ![]() That is because the total number of permutations is just the TOTAL (not just of a single group) arrangements of things (order matters). This formula is intuitive because the total number of permutations (order matters) = total number of ways to arrange the things in a single group (order matters) x the number of groups or combinations. Number of combinations or groups = (total number of permutations )/(total number of ways to arrange the things in a single group ). We use combinations because there is no mention about the position (for example: leader, et cetera), so we assume that they don't care about which seat that a person sit on (order does not matter). Let's take our total permutations and calculate the total number of groups of (in this case) 3, for which we call them the combination because combination is essentially a way to find out how many combinations/groups there are. 1716 is our total number of permutations. Now we've got 156 x 11 = 1716 total scenarios (and we care about the order because this is permutation). Then, each of the the 156 scenarios branches into 11 branches because those are the possible third outcomes for a given second outcome. So far, we have 13 x 12 = 156 different scenarios. I recommend you to visualize it with trees and branches.įirst, we have to understand that 13 x 12 x 11 means that there are 13 initial scenarios that each have 12 branches because those are possible second outcomes for a given first outcome. In other words it is now like the pool balls question, but with slightly changed numbers.To make it more intuitive, I have a more elaborate (I guess) explanation that I've come up myself. ![]() This is like saying "we have r + (n−1) pool balls and want to choose r of them". So (being general here) there are r + (n−1) positions, and we want to choose r of them to have circles. Notice that there are always 3 circles (3 scoops of ice cream) and 4 arrows (we need to move 4 times to go from the 1st to 5th container). So instead of worrying about different flavors, we have a simpler question: "how many different ways can we arrange arrows and circles?" Let's use letters for the flavors: (one of banana, two of vanilla): Let us say there are five flavors of icecream: banana, chocolate, lemon, strawberry and vanilla.
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