Using 5C2 to calculate number of ways to get 2 heads in 5 coin tosses

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In summary, the conversation discusses the use of 5C2 to calculate the number of ways to get two heads from five coin tosses. The explanation given is that the numbers represent the number of choices for each spot, with the permutations needing to be divided by two to avoid double counting. The conversation also mentions that this can be represented as 5x4/2 or as 5!/(2!3!). The final explanation is that this is equivalent to drawing two numbers from a pot of five numbers without replacement, where the order does not matter.
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Homework Statement
How many ways can you get two heads in five coin tosses
Relevant Equations
5C2 = 10
Hi everyone

I'm looking at binomial distributions and one of the examples uses 5C2 to calculate the number of ways you can get two heads from five coin tosses. I don't question that the formula is correct, I just find it hard to picture what the numbers represent.

The numbers make sense to me in the context of 5 choose 2. For example, if you are choosing a team of two from five people, 5 represents the number of choices for the first spot. Four represents the number of choices for the second spot. You then divide the permutations by two to avoid double counting.

What do the numbers mean in the case of five coin flips? I'm picturing it more as a case of 2 choose 5 (even though that's probably not a thing). You can choose between heads or tails and have five flips to fill.

If I were to represent the coin flip situation as 5x4/2, what would the 5, 4 and 2 mean?Thanks
 
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  • #2
HHTTT
HTHTT
HTTHT
HTTTH
THHTT
THTHT
THTTH
TTHHT
TTHTH
TTTHH
 
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Actually, I think I get it now. Would this be a good way to explain it?

If you look at the permutations, a head can occur on any of the five flips. Once you know a head has occurred at a given flip, there are only four other possible places for the second head to occur. The permutations need to be divided by two to avoid double counting. Thus, 5x4/2.
 
  • #4
Darkmisc said:
Actually, I think I get it now. Would this be a good way to explain it?

If you look at the permutations, a head can occur on any of the five flips. Once you know a head has occurred at a given flip, there are only four other possible places for the second head to occur. The permutations need to be divided by two to avoid double counting. Thus, 5x4/2.
That's the idea. You could code those ten combinations (note these are not permutations) by specifying the two places where the heads are. So, HHTTT becomes 12; and TTHTH becomes 35 etc.

And now this is equivalent to drawing two numbers from a pot of five numbers (without replacement) and where the order you draw the numbers does not matter. I.e. drawing number 1 then number 2 is the same as drawing number 2 then number 1. And that is more explicity ##\frac{5 \times 4}{2}##.
 
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  • #5
Darkmisc said:
If I were to represent the coin flip situation as 5x4/2, what would the 5, 4 and 2 mean?
I would not try to understand it that way. I would understand it in the form 5!/(2!3!),
The 5! is the number of ways to order 5 objects if they are all different.
The 2! is the number of ways that the two heads have been over-counted. Swapping the two heads should not be counted twice, so divide by 2!.
The 3! is the number of ways that the three tails have been over-counted. Swapping any of them should not be counted again, so divide by 3!.
 
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FAQ: Using 5C2 to calculate number of ways to get 2 heads in 5 coin tosses

1. How does the 5C2 formula work to calculate the number of ways to get 2 heads in 5 coin tosses?

The 5C2 formula, also known as the combination formula, calculates the number of ways to choose 2 items from a set of 5 items. In the context of coin tosses, this means the number of ways to get 2 heads in 5 coin tosses. The formula is: 5C2 = 5! / (2! * (5-2)!) = 10. This means that there are 10 possible combinations of 2 heads in 5 coin tosses.

2. Why is the 5C2 formula used specifically for calculating the number of ways to get 2 heads in 5 coin tosses?

The 5C2 formula is used because it is a combination formula, which is used to calculate the number of ways to choose a certain number of items from a larger set. In this case, we are choosing 2 heads from a set of 5 coin tosses. Other formulas, such as the permutation formula, are used for different types of counting problems.

3. Can the 5C2 formula be used for other types of counting problems?

Yes, the 5C2 formula can be used for any problem where you need to choose a certain number of items from a larger set. For example, it can be used to calculate the number of ways to choose 2 out of 5 people for a project, or the number of ways to choose 2 out of 5 flavors for an ice cream cone.

4. Is the 5C2 formula the only way to calculate the number of ways to get 2 heads in 5 coin tosses?

No, there are other methods that can be used to calculate this number. For example, you could list out all the possible combinations of 2 heads in 5 coin tosses and count them manually. However, this can be time-consuming and prone to errors, especially for larger numbers. The 5C2 formula is a more efficient and accurate method for calculating this number.

5. What other factors should be considered when using the 5C2 formula to calculate the number of ways to get 2 heads in 5 coin tosses?

When using the 5C2 formula, it is important to consider if the items being chosen are ordered or unordered. In this case, the coin tosses are unordered, meaning the order in which the heads appear does not matter. If the coin tosses were ordered, meaning the first and second toss had to be heads in that specific order, a different formula would need to be used. It is also important to make sure that the formula is being applied correctly and that the numbers used are accurate.

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