Can You Help Me Count the Different Ways to Organize My List?

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In summary, Veronica1999 tried to make an organized list of songs that she likes, but she kept on messing up. She then tried to subtract the cases that don't work, but this also was not a good approach. Veronica1999 asked for help setting up the cases she should be considering.
  • #1
veronica1999
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First I tried to make an organized list but I kept on messing up.
Then I tried to subtract the cases that don't work but this also was not a good approach.
Could I get some help on setting up the cases I should be considering?
 

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  • #2
veronica1999 said:
First I tried to make an organized list but I kept on messing up.
Then I tried to subtract the cases that don't work but this also was not a good approach.
Could I get some help on setting up the cases I should be considering?

Hi veronica1999, :)

Under the given constraints each girl should have 2 or 3 songs that she likes. If a girl likes only one of the songs, then the following condition cannot be satisfied.

For each of the three pairs of the girls, there is at least one song liked by those two girls but disliked by the third.

Similarly, if a girl likes all four of the songs, then the condition,

No song is liked by all three.

cannot be satisfied.

Let me name the girls as A, B and C. Now consider each case,

No. of songs liked by ANo. of songs liked by BNo. of songs liked by C
1)332
2)323
3)233
4)322
5)232
6)223
7)222

If we consider the first case,

'A' likes 3 songs out of 4. There are, \({}^4C_{3}\) ways to choose these three songs. 'B' also likes 3 songs out of 4. Suppose 'B' likes the same three songs that 'A' likes. Then 'C' should like a song that both 'A' and 'B' like. This cannot happen as it is given that,

No song is liked by all three.

Therefore, 'B' should like only two songs that 'A' like, and the other one is the one that 'A' dislikes. The number of ways to choose the two songs(out of the 3 that A likes) is given by, \({}^3C_{2}\). Now if you think carefully you will see that only a pair of songs are left for 'C' to like, without violating the given criteria.

Therefore the total number of possibilities for the first case \(={}^4C_{3}\times{}^3C_{2}\)

If the above explanation is hard to visualize the diagram that I have attached may help. The fours songs are denoted by 1,2,3 and 4.

Likewise I have considered each case separately. These are given in the following table.
No. of songs liked by ANo. of songs liked by BNo. of songs liked by CNo. of Ways to choose the songs
1)332\({}^4C_{3}\times{}^3C_{2}\)
2)323\({}^4C_{3}\times{}^3C_{1}\)
3)233\({}^4C_{2}\times{}^2C_{1}\)
4)322\({}^4C_{3}\times{}^3C_{1}\times{}^2C_{1}\)
5)232\({}^4C_{2}\times{}^2C_{1}\times{}^2C_{1}\)
6)223\({}^4C_{2}\times{}^2C_{1}\times{}^2C_{1}\)
7)222\({}^4C_{2}\times{}^2C_{1}\times{}^2C_{1}\)

Therefore the total number of different ways \(=\left({}^4C_{3}\times{}^3C_{2}\right)+\left({}^4C_{3}\times{}^3C_{1}\right)+\left({}^4C_{2}\times{}^2C_{1}\right)+\left({}^4C_{3}\times{}^3C_{1} \times{}^2C_{1}\right)+3\left({}^4C_{2}\times{}^2C_{1}\times{}^2C_{1}\right)=132\)

Kind Regards,
Sudharaka.
25035mh.png

 
Last edited:
  • #3
Thank you!
You are really awesome.:D
 

What is "counting different ways"?

"Counting different ways" is a mathematical concept that refers to the process of determining the number of ways in which a specific event or outcome can occur.

Why is "counting different ways" important in science?

In science, "counting different ways" is important because it allows us to quantify and analyze the various possible outcomes of an experiment or observation. This can help us make predictions, draw conclusions, and understand the underlying patterns and relationships in our data.

What are some common methods for "counting different ways"?

Some common methods for "counting different ways" include using combinations, permutations, and probability calculations. Other techniques such as tree diagrams, tables, and charts can also be used to visualize and organize the different ways an event can occur.

How does "counting different ways" relate to probability?

Probability is the likelihood of a particular event occurring. "Counting different ways" is closely related to probability as it helps us determine the total number of possible outcomes, which is essential for calculating the probability of a specific event.

Can "counting different ways" be applied to real-world situations?

Yes, "counting different ways" can be applied to real-world situations in various fields such as economics, biology, and physics. For example, it can be used to analyze the possible outcomes of stock market fluctuations, genetic inheritance patterns, or particle collisions in a particle accelerator.

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