00:01
So we're given two questions that really deal with probability.
00:04
The first question has to do with students being selected for a student council position.
00:11
There are 11 total students in the student council activity.
00:14
And the first, you have three, you and your three friends are on this council or part of the 11 students.
00:20
And so the first question says, what's the probability that you and both of your friends are picked by the teacher? they don't specify that order matters in this case.
00:30
So what we you want to do is figure out how many different combinations there are.
00:34
And so we're using the combination formula, which is n factorial divided by n minus r factorial times r factorial.
00:42
So what this is going to do is tell us how many possible combinations there are.
00:47
And of those combinations, how many of them would we expect for all three of these individuals to be selected? so order doesn't matter in this case.
00:57
The order in which we select the three people doesn't matter.
01:00
So that's why we're using the combinations formula.
01:03
So n is the number of items to select from.
01:06
So we have 11 students.
01:07
And then r is the number of items that we are choosing or people that we're choosing.
01:12
We're choosing three at a time.
01:14
So to expand this, we would have 11 factorial over 11 minus r or 11 minus three factorial times three factorial.
01:25
Factorial means it's the number times every number that comes before it.
01:28
So 11 times 10 times nine and so on.
01:31
If i simplify this piece, this becomes eight, so eight factorial.
01:35
So we can think about this numerator, 11 times 9 or times 10 times 9 times 8 and so on, it's going to cancel this 8 factorial.
01:44
If we divide by 8 factorial, it's going to cancel it out and leave us with 11 times 10 times 9.
01:50
The 8 and so on is going to cancel itself out.
01:52
And then we still have this 3 times 2 times 1 in the denominator.
01:57
And so if i multiply, i get 990 divided by 6.
02:01
Which gives me 165.
02:04
So there are 165 total possible combinations.
02:07
And of those combinations, we would expect that one in 165 would choose all three people.
02:15
The second question says that the first person who's chosen becomes the president, the second person chosen is the vice president, and the third person is the treasurer.
02:24
So in this case, order does matter.
02:26
And so what we need to do is consider, well, what happens, you know, when one person is chosen, they're no longer of available to be chosen for the second position.
02:35
So we still have 11 as our n value and three as our r value.
02:41
So it's 11 factorial over 11 minus 3 factorial.
02:47
So notice the really only difference is we're not multiplying by three factorial because we don't have a variety of different combinations of those three people.
02:57
Each one is only being selected once.
02:59
So 11, this again will become eight factorial.
03:03
So it's going to cancel out the 8 times 7, times 6 and so on, which is just going to leave us with 11 times 10 times 9, which is 990...