Solving problems about selections using combinations

In summary, the conversation is about solving problems involving selecting things, specifically finding the number of ways to select a team of 3 men and 2 women from a group of 6 men and 5 women. The concept of permutations and combinations is discussed, with the explanation that permutations are used when order is important and combinations are used when order is not important. The conversation ends with a suggestion to use the methods discussed to find the total number of ways the combined team can be chosen.
  • #1
RigidBody
12
0
hello

i need help solving problems involving selecting things. like for example find the number of ways in which a team of 3 men and 2 women can be selected from a group of 6 men and 5 women.

:bugeye: i know how to do perms and combs but just don't know how to apply them.:eek:

help muchos appreciated
 
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  • #2
Alright, first I'll explain when to use permutations so that combinations make more sense. Suppose you want to know how many 5 letter words there are with no letter repeated. You can think of choosing the letters in order. For the first letter there would be 26 choices. For the second there would only be 25 because you can't use the first letter again. And so on. The number of possibilities is 26*25*24*23*22=[itex]\frac{26!}{21!}[/itex]= 26P5. Use permutations when you are counting the number of ways to choose objects without replacement where the order is important.

Now suppose the order is not important, as when choosing the members of a team. Suppose you are choosing a team of 3 people from a group of twelve. We already figured out that if the order did matter then the number of possibilities would be 12P3. But the order doesn't matter. How do we account for that? Suppose we have a team A,B,C. This team is the same as B,C,A or A,C,B and so on. From what I said in the first paragraph, its clear that the number of teams consisting of A,B and C is 3P3=[itex]\frac{3!}{(3-3)!}=\frac{3!}{0!}=3![/itex]. So each team is counted 3!=6 times. This means that the real number of teams is only one sixth of what we counted. So the real number of teams where the order does not matter is 12P3/3!=[tex]\frac{\frac{12!}{(12-3)!}}{3!}=\frac{12!}{(12-3)!3!}[/tex]=12C3. Use combinations when choosing things without replacement when order does not matter.

Finally, coming to your example. Basically you can think of what you are doing as choosing two teams: one of women and one of men. You can use the method I just decribed to find the number of ways each of those teams could be chosen. Then how do you find the total number of ways the combined team could be chosen? See if you can reason it out.
 
  • #3


Hi there,

No problem, I can definitely help you with solving these types of problems using combinations. First, let's review what combinations are. Combinations are used when we want to select a group of items from a larger set, where the order of selection does not matter. In your example, we want to select a team of 3 men and 2 women from a group of 6 men and 5 women. This means that the order in which we select the team members does not matter, as long as we end up with the required number of men and women.

To solve this problem, we can use the combination formula: nCr = n!/r!(n-r)!, where n is the total number of items and r is the number we want to select. In this case, n = 11 (6 men + 5 women) and r = 5 (3 men + 2 women).

Plugging in these values, we get: 11C5 = 11!/5!(11-5)! = (11*10*9*8*7)/(5*4*3*2*1) = 462.

So, there are 462 ways to select a team of 3 men and 2 women from a group of 6 men and 5 women. I hope this helps and let me know if you have any other questions. Good luck!
 

What is a combination?

A combination is a selection of objects from a larger group without regard to their order. For example, if you have a set of 10 marbles and you want to pick 3 of them, the different ways you can pick those 3 marbles is a combination.

What is the formula for finding the number of combinations?

The formula for finding the number of combinations is nCr = n! / (r!(n-r)!), where n is the total number of objects and r is the number of objects to be selected.

How do I solve problems involving combinations?

To solve problems involving combinations, you first need to identify the given information, such as the total number of objects and the number of objects to be selected. Then, use the combination formula to calculate the number of combinations. Finally, use the number of combinations to solve the specific problem.

Can combinations be used in real-life situations?

Yes, combinations can be used in various real-life situations, such as in probability and statistics, genetics, and computer science. For example, combinations can be used to calculate the number of possible outcomes in a game of chance or to analyze gene combinations in biology.

What are some common mistakes when working with combinations?

Some common mistakes when working with combinations include using the wrong formula, not considering the order of objects, and not accounting for repeated objects. It is important to carefully read and understand the problem and double check your calculations to avoid these mistakes.

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