# How to find r in the equation P(5,r)=20

• MiniTank
In summary, the conversation discusses solving for a variable, r, in the equation P(5,r)=20 using algebraic techniques. The person asking the question suggests using factorial notation and the other person provides helpful hints on how to approach the problem. The conversation then moves on to a similar question involving permutations and combinations, with the second person providing hints on how to approach the problem. The final solution is given as 5184 for the second question.
MiniTank
how would you solve for r in this situation:

P(5, r) = 20

Now I understand you can't use trial and error to find that r=2 but is there any other way using an algebraic technique. Itried something like this..
$$\frac{5!}{(5-r)!}=20$$
then
$$\frac{5\times4\times3\times2\times1}{20}=(5-r)!$$
$$6=(5-r)!$$

now i don't know what to do next

Express 6 as a factorial.

And consider what $$(5-r)!$$ means.

$$(5-r)! = (5-r)*(5-r-1)*(5-r-2)$$ etc...

ahh i see then you get 3!=(5-r)!, then cancel out the factorials and you get 3=5-r.. solve for r .. r=2 .. thanks, that's been bothering me alot

i got another question but I am not sure if it has to do with permutations or but it has a similar subject. should i post it in this topic?

Go ahead. Also I didnt check your work on the first one, but if all is correct til the last step then r = 2.

a) In how many ways can they be sated in a row if people of the same nationality are to be seated next to each other?

b) redo (a) if they sit at a round table

for a) the way i look at it is for the Canadians there are three ways which they can be seated so you get 3!, then same for the americans and mexicans getting 4! and 2! respectively. Then you have a total of three nationalities which means they can be switched around giving you 3!.. Therefore in total you get

3!(3!+4!+2!)
=3!(32)
=192

not sure if that's right

b)im pretty sure this is wrong but here's what I am thinking
$$\frac{8!}{3!4!2!} =4\times7\times5 =140$$

any takers?

k scratch what i did for (b) i just figured out that's wrong.. assuming what i did in (a) was right, then (b) should be something like
$$1\bullet2!(1\bullet2!+1\bullet3! +1\bullet1!)$$
$$=2(9) = 18$$

MiniTank said:

a) In how many ways can they be sated in a row if people of the same nationality are to be seated next to each other?

b) redo (a) if they sit at a round table

for a) the way i look at it is for the Canadians there are three ways which they can be seated so you get 3!, then same for the americans and mexicans getting 4! and 2! respectively. Then you have a total of three nationalities which means they can be switched around giving you 3!.. Therefore in total you get

3!(3!+4!+2!)
=3!(32)
=192

not sure if that's right

b)im pretty sure this is wrong but here's what I am thinking
$$\frac{8!}{3!4!2!} =4\times7\times5 =140$$
SOLUTION HINTS:
{# in C Block} = 3
{# in A Block} = 4
{# in M Block} = 2

a)
There are (3!) arrangements of the 3 Block Units. For each Block Unit arrangement, there are (3!) sub-arrangements within the "C" Block, (4!) sub-arrangements within the "A" Block, and (2!) sub-arrangements within the "M" Block.
{Total Arrangements} = (3!)*{(3!)*(4!)*(2!)}

b)
The round table has 9 chairs. Let the chair closest to North direction be designated "N". Chair "N" will be occupied by a member of EITHER "C", "A", or "M". If it's "C", there are 3 possible Block positions relative to "N" (0 members right of "N", 1 member right of "N", & 2 members right of "N"). For each relative position, there are (3!) sub-arrangements. Further, the 2 remaining Blocks ("A" & "M") have 2 possible relative positions (to left of "N" or to right of "N"), with (4!) sub-arrangements within "A" Block and (2!) sub-arrangements within "M" Block. Thus, total arrangements possible when "C" Block occupies chair "N" is:
{(3)*(3!)}*(2)*{(4!)*(2!)} ::: <--- "C" occupies "N"
Calculate similar results for cases when Blocks "A" & "M" occupy chair "N", and add all three results together for total number of arrangements:
{(3)*(3!)}*(2)*{(4!)*(2!)} + ? + ?

~~

xanthym said:
SOLUTION HINTS:
{# in C Block} = 3
{# in A Block} = 4
{# in M Block} = 2

a)
There are (3!) arrangements of the 3 Block Units. For each Block Unit arrangement, there are (3!) sub-arrangements within the "C" Block, (4!) sub-arrangements within the "A" Block, and (2!) sub-arrangements within the "M" Block.
{Total Arrangements} = (3!)*{(3!)*(4!)*(2!)}

b)
The round table has 9 chairs. Let the chair closest to North direction be designated "N". Chair "N" will be occupied by a member of EITHER "C", "A", or "M". If it's "C", there are 3 possible Block positions relative to "N" (0 members right of "N", 1 member right of "N", & 2 members right of "N"). For each relative position, there are (3!) sub-arrangements. Further, the 2 remaining Blocks ("A" & "M") have 2 possible relative positions (to left of "N" or to right of "N"), with (4!) sub-arrangements within "A" Block and (2!) sub-arrangements within "M" Block. Thus, total arrangements possible when "C" Block occupies chair "N" is:
{(3)*(3!)}*(2)*{(4!)*(2!)} ::: <--- "C" occupies "N"
Calculate similar results for cases when Blocks "A" & "M" occupy chair "N", and add all three results together for total number of arrangements:
{(3)*(3!)}*(2)*{(4!)*(2!)} + ? + ?

~~

so for b) you get

[3(3!)][2(4!2!)] + [2(2!)][2(3!4!] + [4(4!)][2(3!2!)]
=5184

Alright thanks

## 1. What is the equation P(5,r)=20?

The equation P(5,r)=20 represents a mathematical relationship between two variables, where the value of r is unknown. P stands for probability and the equation is asking for the value of r that would make the probability equal to 20 when the other variable is 5.

## 2. How do we find the value of r in this equation?

To find the value of r, we need to use algebraic methods such as substitution or elimination. We can also use a graphing calculator or a spreadsheet to solve the equation.

## 3. What information do we need to solve this equation?

In order to solve this equation, we need to know the values of the other variable (in this case, 5) and the probability value (20). We also need to have a basic understanding of algebraic concepts and equations.

## 4. Can this equation have more than one solution for r?

Yes, this equation can have multiple solutions for r. This means that there can be more than one value of r that would make the probability equal to 20 when the other variable is 5.

## 5. What does the solution for r represent in this equation?

The solution for r represents the value of the variable that would make the probability equal to 20 when the other variable is 5. In other words, it is the value of r that satisfies the equation P(5,r)=20.

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