Applied Mathematics question on Permutation

[Imilling139]

Hi! I’m not sure if what I’m studying is advanced math or not but I didn’t see a category for applied math so I’m just gonna post in here. I’m currently writing notes of permutations and I do not understand something. I will attach the problem and the context so that someone can help me the best way they can. Okay so I understand everything up until the line following the statement “we can write this answer as a permutation”. It starts with 20=5*4= ....... I do not understand why we used the those specific numbers in the fraction. The numerator makes sense because it starts with 5 and then does *4*3*2*1 because of the 5 in 20=5*4. But the denominator does not start with 4??? It’s just skips it and goes straight to 3*2*1?? I’m sorry if this is a little chaotic but I am literally failing math right now and I really need help. If there is any more clarification that is needed to accurately give me answers just let me know and I’ll do it! Thanks so much.

 

[pka]

It is a simple rule: If \(N\ge j\) then \(_N\mathcal{P}_j=\dfrac{N!}{(N-j)!}\).
So \(_{12}\mathcal{P}_3=\)\(\dfrac{12!}{9!}\)

 

[Imilling139]

It is a simple rule: If \(N\ge j\) then \(_N\mathcal{P}_j=\dfrac{N!}{(N-j)!}\).
So \(_{12}\mathcal{P}_3=\)\(\dfrac{12!}{9!}\)

Hi! Thank you for replying! I understand the eventual end to the problem but I do not understand the steps required to get there... can you explain?

 

[JeffM]

It might help if you understood the intuition behind the rule given by pka.

Suppose we have 7 distinct objects, and we want to know how many ways we can select the first one. Obviously

[MATH]7 = 7 * 1 = 7 * \dfrac{(7 - 1)!}{(7 - 1)!} = \dfrac{7 * 6!}{(7-1)!} = \dfrac{7!}{(7-1)!}.[/MATH]
How many ways can we select first one, then a second?

There are 7 ways to select the first and 6 ways to select the second.

[MATH]7 * 6 = 7 * 6 * 1 = 7 * 6 * \dfrac{5!}{5!} = \dfrac{7 * 6 * 5!}{(7-2)!} = \dfrac{7!}{(7-2)!}.[/MATH]
How many ways can we select first one, then a second one, then a third one? Well, it is

[MATH]7 * 6 * 5 = 7 * 6 * 5 * 1 = 7 * 6 * 5 * \dfrac{4!}{4!} = \dfrac{7 * 6 * 5 * 4!}{(7 - 3)!} = \dfrac{7!}{(7-3)!}.[/MATH]
Generalizing from that, the number of distinct ways you can order j distinct items selected from n distinct items is

[MATH]\dfrac{n!}{(n - j)!}.[/MATH]
Formulas stick in your memory better when they make sense, and you are less likely to use the wrong formula if you understand why the different formulas work.

 

[Imilling139]

It might help if you understood the intuition behind the rule given by pka.

Suppose we have 7 distinct objects, and we want to know how many ways we can select the first one. Obviously

[MATH]7 = 7 * 1 = 7 * \dfrac{(7 - 1)!}{(7 - 1)!} = \dfrac{7 * 6!}{(7-1)!} = \dfrac{7!}{(7-1)!}.[/MATH]
How many ways can we select first one, then a second?

There are 7 ways to select the first and 6 ways to select the second.

[MATH]7 * 6 = 7 * 6 * 1 = 7 * 6 * \dfrac{5!}{5!} = \dfrac{7 * 6 * 5!}{(7-2)!} = \dfrac{7!}{(7-2)!}.[/MATH]
How many ways can we select first one, then a second one, then a third one? Well, it is

[MATH]7 * 6 * 5 = 7 * 6 * 5 * 1 = 7 * 6 * 5 * \dfrac{4!}{4!} = \dfrac{7 * 6 * 5 * 4!}{(7 - 3)!} = \dfrac{7!}{(7-3)!}.[/MATH]
Generalizing from that, the number of distinct ways you can order j distinct items selected from n distinct items is

[MATH]\dfrac{n!}{(n - j)!}.[/MATH]
Formulas stick in your memory better when they make sense, and you are less likely to use the wrong formula if you understand why the different formulas work.

This makes so much sense, thank you so much!!

 

[Dr.Peterson]

Okay so I understand everything up until the line following the statement “we can write this answer as a permutation”. It starts with 20=5*4= ....... I do not understand why we used the those specific numbers in the fraction. The numerator makes sense because it starts with 5 and then does *4*3*2*1 because of the 5 in 20=5*4. But the denominator does not start with 4??? It’s just skips it and goes straight to 3*2*1??

It may also help to understand the thinking behind each piece of that line:

First, they have the calculation they just did, 5*4 = 20. That alone is all you need to answer the question. But then they are showing other ways to think of that calculation that can be generalized. We could just describe it as "start with 5, and multiply by successive numbers counting down for 2 numbers", but that's a little awkward.

Next, they multiply the numerator and denominator by 3*2*1, which doesn't change the value, but makes it look pretty! That is, it shows a pattern that can be useful; it's really just a different way to say the same thing, but leads in the direction of using factorials. It sort of says, start at 5, and multiply all the way down to 1, but ... don't include the numbers starting at 3.

The next thing they show expresses this explicitly, as 5!/3!. Why do this? Just because we already have a notation for multiplying all the way down to 1, and want to reuse it. Too many different notations makes things complicated.

Then they start to generalize: the 5 is the number of items to choose from; where did the 3 on the bottom come from? That's 2 less than the 5, and the 2 is the number of slots to put items in. So 5!/(5-2)! expresses the calculation directly in terms of the two numbers in the problem.

Finally, they just give a name to this thing we've calculated, "the number of permutations of 5 items taken 2 at a time", the symbol \(_5P_2\).

All the expressions on that line are different ways to say the same thing, and each is useful in different places. The notation at the end is useful because calculators often have a button for that, which can do complicated calculations you don't want to have to write out!

After all this, they define that last notation in general, using the form they'd ended up with:

This gives us a simple way to tell someone to do that calculation, which, ultimately, just means "start with n, and multiply by successive numbers counting down for r numbers" -- but not so wordy.

 

[Imilling139]

It may also help to understand the thinking behind each piece of that line:
View attachment 25795
First, they have the calculation they just did, 5*4 = 20. That alone is all you need to answer the question. But then they are showing other ways to think of that calculation that can be generalized. We could just describe it as "start with 5, and multiply by successive numbers counting down for 2 numbers", but that's a little awkward.

Next, they multiply the numerator and denominator by 3*2*1, which doesn't change the value, but makes it look pretty! That is, it shows a pattern that can be useful; it's really just a different way to say the same thing, but leads in the direction of using factorials. It sort of says, start at 5, and multiply all the way down to 1, but ... don't include the numbers starting at 3.

The next thing they show expresses this explicitly, as 5!/3!. Why do this? Just because we already have a notation for multiplying all the way down to 1, and want to reuse it. Too many different notations makes things complicated.

Then they start to generalize: the 5 is the number of items to choose from; where did the 3 on the bottom come from? That's 2 less than the 5, and the 2 is the number of slots to put items in. So 5!/(5-2)! expresses the calculation directly in terms of the two numbers in the problem.

Finally, they just give a name to this thing we've calculated, "the number of permutations of 5 items taken 2 at a time", the symbol \(_5P_2\).

All the expressions on that line are different ways to say the same thing, and each is useful in different places. The notation at the end is useful because calculators often have a button for that, which can do complicated calculations you don't want to have to write out!

After all this, they define that last notation in general, using the form they'd ended up with:
View attachment 25797
This gives us a simple way to tell someone to do that calculation, which, ultimately, just means "start with n, and multiply by successive numbers counting down for r numbers" -- but not so wordy.

Thank you so much, this is amazing! ?