# Number of ways to arrange 9 people in 5 spots, given conditions

1. Dec 5, 2015

### leo255

1. The problem statement, all variables and given/known data

Nine people (Ann, Ben, Cal, Dot, Ed, Fran, Gail, Hal, and Ida) are in a room. Five of them stand in a row for a picture. In how many ways can this be done if

(e) Hal or Ida (but not both) are in the picture?

(f) Ed and Gail are in the picture, standing next to each other?
1. (g) Ann and Ben are in the picture, but not standing next to each other?
2. Relevant equations

3. The attempt at a solution

Given no conditions, the answer would obviously be P(9, 5).

For part "e," one of them can be in the picture. Take both out of the entire pool, and one out of the group, taking pictures. This gives me P(7, 4).

For part "f," I can take two out of both groups (i.e. total and group picture), that leaves me P(7, 3).

For part "g," I don't know how to calculate how they will be ordered, other than to give an incomplete answer of P(7, 3).

I know that all of my answers are wrong.

2. Dec 5, 2015

### Ray Vickson

How do you know you answers are all wrong; what do you think are the correct answers?

Anyway, what do you want us to do? Confirm your suspicions? Show you how to do the questions (which would violate PF rules)?

3. Dec 5, 2015

### geoffrey159

If the notation P(a,b) is the number of combinations of b elements among a, then your answers are incorrect but not stupid.
You forgot to account for a number of configurations in each case. So the correct answers are a multiple of your answers.

Last edited: Dec 5, 2015
4. Dec 5, 2015

### leo255

I actually have answers, but they do me no good if I can't figure this on my own. I honestly try not to look at the answers, but here they are:

e is 2 * 5 * P(7, 4)
f is 2 * 4 * P(7, 3)
g is 5 * 4 * P(7,3) - 2 * 4 * P(7, 3)

I obviously got one part of them correct. To be honest, I probably don't even need help with g, since it just takes from f (I understand what 5 * 4 * P(7,3) means), but I am really confused about e and f, and how these numbers are gotten.

Specifically for e, if only one person is in the picture, I could see 5 * P(7, 4), but I don't know why we would need a 2. I am clueless about where the 2 * 4 comes from for f.

Last edited: Dec 5, 2015
5. Dec 5, 2015

### leo255

Ok, so answering my own question, I now see where the 2 comes from for e. We need to account for the possibility of one out of the two being in the picture. Still working on f.

6. Dec 5, 2015

### Ray Vickson

For the first one, the total number of arrangements = arrangements that have Ida but not Hal + arrangements that have Hal but not Ida. These two numbers are the same, and the first one = 5*P(7,4). Note that adding the two numbers together is OK (no "double counting", for example) because the total number of pictures is the number containing Ida plus the number containing Hal. For the second one, let E and G form a new "superperson", like two persons glued together. You need to locate that one superperson plus three others from the remaining 7; the remaining three can be arranged in P(7,3) ways, then the superperson inserted in 4 ways, then be separated again into E and G in two possible orders.
The last one looks at the number of pictures having both A and B in them, then subtracts the number where A and B are next to each other.

Last edited: Dec 5, 2015
7. Dec 5, 2015

### leo255

Thanks! I think I got it. I was really getting confused by the '4,' but see that those are the only possibilities (i.e. 1,2; 2,3; 3, 4; 4, 5) - the fifth option would be 5, 6, and that would not exist in this context. The 2, as you said, is just where they happen to be positioned.

8. Dec 5, 2015

### geoffrey159

Let's take (e) :
P(7,4) is the number of sets of 5 person in the pool containing Hal but not Ida (or Ida but not Hal). In each of these sets, people can be ordered in $5!$ different ways. So the answer should be $2\times (5!)\times P(7,4)$ and not $2\times 5\times P(7,4)$.

9. Dec 5, 2015

### Ray Vickson

The notation $P(n,m)$ means the number of permutations of $m$ things chosen from a group of $n$, so already includes all the different "orders". In contrast, the number of combinations of $m$ things chosen from $n$ things is $C(n,m) = P(n,m)/m!$, because when disregarding the order the same $m$ things can be ordered in $m!$ different ways.

So, $2 \times 5 \times P(7,4)$ is correct, because for each of the $P(7,4)$ permutations of the 4 things we can insert the 5th thing into 5 different places.

10. Dec 5, 2015

### geoffrey159

Oh Ok, I asked the the OP what was the meaning of this notation but did not reply about that so I assumed it was the number of combinations. So I agree then !

11. Dec 5, 2015

### leo255

Sorry for not making all of that clear. Again, thanks guys for your help!