Permutations of taking 2 letters from (a, a, b, c)

[jaus tail]

Homework Statement

All permutations made with letters (a, a, b, c) taken two at a time are:

Homework Equations

Permutation is ⁿP_r where n is total number of things, r is number of things taken at a time.
If P₁ objects are identical, P₂ objects are identical such that P₁ + P₂ = n
then permutation is
n!/(P₁!P₂ !)

The Attempt at a Solution

Here letters are (a, a, b, c)
Four letters. n = 4.
Taking 2 at a time. means r = 2

But two letters repeat so P₁ = 2.
letter b comes once. so P₂ = 1
Letter c comes once so P₃ = 1

Permutation is: 4!/(2!)(1!)(1!) = 4 * 3*2/(2) = 12

But if i arrange the letters I get:
(a, a, b, c) taking 2 letters at a time:

1. aa
2. ab
3. ba
4. ac
5. ca
6. bc
7. cb

I get 7 combinations. Where am I wrong?
Whereas by formula I get answer as 12. (above in italics)

PS ( I know I'm posting a lot. But i have a big exam of math and electronics on 11th feb and thus am revising all topics together. )

 

[Orodruin]

You are applying an argumentation that does not apply. $n!/(p_1!\ldots p_k!)$ has no direct relation to what you want to compute, it is the number of different unordered ways of placing $n$ objects into $k$ bins such that bin $k$ contains $p_k$ objects.

 

[Orodruin]

Also, 4!/(2!) = 4*3*2/2 = 12.

 

[jaus tail]

Hmm... thanks for pointing out my mistake. I've rectified it now.
Though honestly I'm still not clear when to use the formula and when to not.
Will read more examples from book and hope that helps..

Thanks...

 

[Orodruin]

12 would be the number of possibilities if your letters were distict. The boxes would be 2 boxes with a single letter (first and second letter, respectively) and one box with two letters (the unused letters). Now, your letters are not distinct so you will need to account fore double counting duplicates.

 

[jaus tail]

Can I put the first 2 letters in separate boxes?
Question says permutations with 2 letters taken at a time from (a, a, b, c).
So i got 2 boxes and i need to put letters from here.
Case can be:
aa, ab, ba, ac, ca, bc, cb

Is there some formula for this?

I tried another case:
letters (a, a, b, c, d)
Permutations with 2 letters is:
aa
ab ba ac ca ad da bc cb bd db cd dc
Total = 13.
But i couldn't come up with any formula.

 

[Orodruin]

I think you should stop focusing on having a formula and instead think about how to reason in order to arrive at the correct result. It will help you in the long run.

 

[jaus tail]

Yeah true. I made a big table with 4 and 5 letters and different repeating values. Wasn't able to derive a formula though.
Thanks.

 

[jaus tail]

Well i learned one thing that the multi set equation (of dividing with P ! when I is number of identical objects) can be used only when all objects are taken.
So some good thing came out of this exercise.