In how many ways can we select a committee of four Republicans, three Democrats, and two Indepen

In how many ways can we select a committee of four...

Key Concepts

Combinations

This concept involves selecting a subset of items from a larger set where the order of selection does not matter. It is fundamental in problems where specific groups or committees need to be formed from a pool of candidates.

Binomial Coefficient

Often represented as 'n choose k', the binomial coefficient calculates the number of ways to choose k distinct items from a set of n items using the formula n!/(k!(n-k)!). It is crucial in determining the number of possible selections in a committee formation problem.

Multiplicative Principle

Also known as the fundamental counting principle, this concept states that when making several independent selections, the total number of outcomes is the product of the number of outcomes for each individual selection. This principle is essential when combining choices for different groups in a committee.

Independent Selection

This concept emphasizes that each selection is made separately from different groups, meaning the choice in one group does not affect the choices in another. It allows the use of the multiplicative principle when computing the total number of ways to form the committee.

Transcript

00:01 In the given question we are told that there is a group of 10 distinct republicans, there is a group of 10 distinct republicans, 10 distinct republicans, 12 distinct democrats, 12 distinct democrats, and four distinct independents.

00:27 So four distinct independents.

00:30 Right so this is the group of people that are given to us and we are asked in how many ways can be select a committee a committee of four republicans four republicans three democrats three democrats and two independents and two independence and two independence so this is what we are told to find the number of ways in which this committee can be formed from the above group so we are going to use the combination formula over here and according to this formula what it uses the number of ways in which number of ways in which are objects are objects can be selected are distinct objects can be selected from can be selected from n distinct objects and distinct objects right so now this is given by the formula n c r and it is evaluated as n factorial divided by r ferole times n minus r factorial so this is the required formula that we are going to use to solve this problem over here.

02:06 So what we would do over here first is we need in the committee, we need four republicans out of a total of 10 republicans, right? and we need three democrats from a total of 12 democrats and two independents from a total of 4 independence in the group.

02:30 So what we can do is we would take this as let's write the combination of choosing four republicans from a group of 10 times the combination of choosing three democrats from a group of 12 times the combination of choosing two independents from a group of four.

02:58 So this is what the answer would look like.

03:03 So we can just write over here the number of ways...

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