From a group of 10 men and 15 women, how many committees of...

From a group of 10 men and 15 women, how many committees of size 9 are possible?
a. With no restrictions
b. With 6 men and 3 women
c. With 5 men and 4 women if a certain man must be on the committee

Key Concepts

Restricted Combinatorics

Restricted combinatorics involves counting the selection of objects under additional constraints or conditions. It addresses scenarios where selections must adhere to specific requirements – for example, ensuring a certain number of elements from distinct subsets or including predefined individuals into a selection.

Counting Principles

Counting principles include strategies like the addition and multiplication rules, which are used to break down complex counting problems into simpler, independent parts. These principles are employed to systematically count outcomes, particularly when different parts of a problem have varying restrictions or conditions.

Binomial Coefficient (Combination Formula)

The binomial coefficient, commonly written as C(n, k) or 'n choose k', calculates the number of ways to choose k items from a set of n items without considering the order. This formula is crucial for problems that entail forming committees or groups where the order of selection does not matter.

Combinatorics

Combinatorics is the branch of mathematics that deals with counting, selection, and arrangement of objects. It introduces theories and techniques for calculating the number of possible arrangements or selections under varying conditions, and is the foundation for solving problems involving committees, groups, and selections.

Transcript

00:01 Okay, so here says we have 10 men, 15 women, which means we have 25 total in this group.

00:06 It says they need to have a committee of nine.

00:09 And it says how many ways can this committee be selected if part a, anyone can be on the committee, right? it doesn't matter if it's all male, all female, or a mix.

00:19 It really shouldn't matter.

00:20 So here, right, since it's anyone, there's no specification, you're just going to say, okay, well, i've got 25 total.

00:27 Oh, so that was pretty way.

00:28 25 total, i'm going to be choosing nine of them, right? any nine, it doesn't really matter.

00:34 And you can plug that into your formula, which is 25 factorial over 25 minus 9 factorial, 9 factorial, or use the programming on your calculator.

00:44 Either way, you should end up with 204, 2975.

00:52 At least that's what i got when i did it.

00:54 So there are about 2 million different ways to pick a committee of 9.

00:59 Can have any kind of combination.

01:02 Now here on part b it says how many different ways can we have six men and then three women beyond the committee.

01:09 And so what this means is we're going to have to kind of break it up.

01:11 So when we're talking about men, right, there's 10 men to choose from...