Probability: Permutations/Combinations

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In summary, the conversation discusses a course with 4 sections and 3 students picking sections at random. The sample space is 64, with the probability of all students ending up in the same section being 3/64, and the probability of all students ending up in different sections being 6/64. The probability of nobody picking section 1 is 6/64. The conversation also mentions repeating this scenario for n sections and s students and asks for help in deriving the number of ways to do each.
  • #1
catbearbig
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I'm having a hard time understanding how to derive the # ways to do things.

(a) A course has 4 sections with no limit on how many can enrol in each section. 3 students each pick a section at random.

(i) Specify Sample Space S (64?)
(ii)Find the probability that they all end up in the same section (3C1 /64 = 3/64??)
(iii) Find the probability that they all end up in different sections (3P3/64 = 6/64 ??)
(iv) Find the probability that nobody picks section 1 ( 3C2 * 2! / 64 = 6/64??)

(b) Repeat (a) in the case where there are n sections and s students.

Can someone please help me derive their answers of how they get the # of ways to do each. Thanks a lot!
 
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  • #2
I suggest, first, that you check the top thread on how to present a homework problem. Or look at the standard layout used by most others.

For your probability answer, don't forget there are 4 sections, not 3. And a little more explanation of your thinking would help others to help you.
 

1. What is the difference between permutations and combinations?

Permutations and combinations are both ways of arranging or selecting objects from a set. However, permutations take into account the order of the objects, while combinations do not. In other words, in permutations, the order of the objects matters, while in combinations, the order does not matter.

2. How do I calculate the number of permutations?

The number of permutations can be calculated by using the formula n! / (n - r)!, where n is the total number of objects and r is the number of objects being selected. For example, if you have 5 objects and you want to select 3 of them, the number of permutations would be 5! / (5 - 3)! = 5! / 2! = 5 * 4 * 3 = 60.

3. What is the formula for calculating combinations?

The formula for calculating combinations is n! / (r! * (n - r)!). Similar to permutations, n represents the total number of objects and r represents the number of objects being selected. For example, if you have 6 objects and you want to select 4 of them, the number of combinations would be 6! / (4! * (6 - 4)!) = 6! / (4! * 2!) = 6 * 5 / 2 = 15.

4. How do I know when to use permutations or combinations?

You should use permutations when the order of the objects is important. For example, if you are arranging a sequence of events or creating a password, the order of the objects matters. You should use combinations when the order of the objects does not matter. For example, if you are selecting a group of people for a committee, the order in which they are selected does not matter.

5. Can permutations and combinations be used in real-life situations?

Yes, permutations and combinations are used in various real-life scenarios, such as in probability and statistics, computer science, and even in everyday decision making. For example, when you are shuffling a deck of cards, you are essentially creating a permutation of the cards. When you are choosing a combination of toppings for your pizza, you are using combinations to determine the number of possible combinations.

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