The presidents, vice presidents, and secretary-treasurers...

The presidents, vice presidents, and secretary-treasurers from each of four classes are eligible for an all-school council. How many ways can four officers be chosen from these representatives? How many ways can they be chosen if the president must be selected from the sitting presidents, the vice president from the sitting vice presidents, the secretary from the sitting secretary-treasurers, and the treasurer from everybody who's left?

Key Concepts

Counting with Restrictions

Counting with restrictions involves scenarios where selections must satisfy specific criteria, such as choosing items from designated subgroups or following particular assignment rules. In these cases, the count is determined by first applying the restricted conditions for each selection and then using principles like the multiplication principle to combine the counts for each step.

Combinations

In combinatorics, combinations are used to count the number of ways to select items from a larger set when the order of selection does not matter. This is typically represented using the binomial coefficient, often written as 'n choose k', which calculates the number of subsets of k items that can be chosen from a set of n items.

Multiplication Principle

The multiplication principle states that if a process involves several independent steps, where the first step can be done in m ways and the second in n ways, then the total number of ways to complete the process is m multiplied by n. It is a fundamental tool in counting that applies to sequential decisions.

Transcript

00:01 Let's start with part one where we are determining the number of ways four officers can be chosen from these representatives.

00:07 There's four classes each with four representatives, which means there's a total of 16 representatives.

00:21 So we get 16, this is our combination problem.

00:25 There's 16 total spots, total representatives, and there's four levels or four classes.

00:31 So we get 16 factorial divided by 16 minus four factorial.

00:38 Times four factorial.

00:40 Now, when we do this, i'm actually going to leave the 16 factorial for just a moment, divided by 16 minus 4 is 12 factorial times 4 factorial.

00:53 Okay.

00:54 So a factorial means we are multiplying it by the number below it.

01:00 16 would be 16 times 15 times 14 times 13 times 12, so on and so forth till we get to one.

01:05 You're multiplying by every number that comes before.

01:09 So if we do this and write this out, 16 times 15 times 14 times 13 times 12 factorial, i'm going to stop there and i'm going to show you why in just a moment, divided by 12 factorial times 4 factorial.

01:22 Remember, anything divided by itself is one.

01:26 That means 12 factorial divided by 12 factorial is going to cancel out.

01:31 And now we're left with 16 times 15 times 14 times 13.

01:36 Over 4 factorial, which is 4 times 3 times 2 times 1.

01:41 All right.

01:42 16 times 15 times 14 times 13 gives me 43 ,680.

01:48 Divided by 4 times 3 times 2 times 1 is 24.

01:53 4 ,4 ,680 divided by 24 gives me a total number of choices of 1 ,820.

02:02 So that is the number of ways that they could be chosen.

02:04 Now let's look at part two.

02:09 In this case, each position representative must be on the same position in their representative class to be eligible.

02:16 The president must be chosen from the current president.

02:19 The vice president must be chosen from the current vice president.

02:23 And the secretary must be chosen from the current secretary treasurer.

02:27 So for the president, we get, there's four the combination and one spot to fill.

02:39 Vice president is the same but now the secretary must be chosen from the current secretary treasurers so now there's eight because there's two spots four secretaries four treasurers so we get eight and we're choosing one since there are three people chosen from the three of the positions the representatives from the position of treasurer will be 16 minus three which is 13 the treasure must be chosen from who's left so the treasurer then is 13 combination for that one spot.

03:28 13 choices, one spot to fill.

03:31 So by fundamental counting rules, we multiply those together.

03:36 And this gives us the president times the vice president, times the secretary, times the treasurer...

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