Math help- permutations and combinations

[DragonFaerie]

DS had a word problem today that involved finding the number of possible combinations for a two-scoop ice cream cone when there were six flavors of ice cream to choose from. I was able to find tutorials on Khan about permutations and combinations and the formulas for each using factorials (the whole _nC_r formula), but this seemed pretty advanced for 5th grade (the word problem is in Evan Moor's Daily Word Problems for 5th grade book). Is there a simpler way to explain this?

[Cosmos]

I would expect elementary students to list all the possible solutions and then count them. Name the flavors A, B, C, etc., or something else easy to write. Hopefully the student would see that one can be methodical about how he lists them and thus be sure not to miss any or overcount.

 

I would definitely NOT show a fifth grader the formula for computing combinations. That is something that can be understood very easily, but only after trying lots of these kinds of problems and understanding the connection between permutations and combinations.

 

If he is interested in this kind of problem, he could list all the combinations that the problem asks for (where the order of the scoops doesn't matter) and also all the permutations (where the order does matter). He'll be able to see exactly why there are twice as many combinations.

[jennynd]

you can do a tree chart.. (i made it up the name, i don't remember how it was called)

you have 6 branches for the 1st flavor , and each brach has another 6 branches if it is allow same flavors for the 2 scoop. so 6*6 you get 36. then u take repeat out???

[Arcadia]

For 5th grade, they are still suppose to draw the probability tree diagram (6 branch, 6 sub-branch from each branch) and/or list all choices

 

AA, AB, AC, AD ,AE ,AF

BA, BB, BC, BD, BE, BF

CA, CB, CC, CD, CE, CF

DA, DB, DC, DD, DE, DF

EA, EB, EC, ED, EE, EF

FA, FB, FC, FD, FE, FF

 

or calculate

(number of choice) x (number of choice) = 6 x 6 = 36

 

Sample of a tree diagram

[regentrude]

should not do math in a hurry

[DragonFaerie]

But 36 is not the right answer. The answer is 21 (because you can have both scoops of the same flavor). And if you use the formula for combinations with repetitions allowed, the answer does come out to be 21. DS got 36 trying it by writing it all out.

[jennynd]

But 36 is not the right answer. The answer is 21 (because you can have both scoops of the same flavor). And if you use the formula for combinations with repetitions allowed, the answer does come out to be 21. DS got 36 trying it by writing it all out.

 

right you take out the repeats that give you 21

[DragonFaerie]

For 5th grade, they are still suppose to draw the probability tree diagram (6 branch, 6 sub-branch from each branch) and/or list all choices

 

AA, AB, AC, AD ,AE ,AF

BA, BB, BC, BD, BE, BF

CA, CB, CC, CD, CE, CF

DA, DB, DC, DD, DE, DF

EA, EB, EC, ED, EE, EF

FA, FB, FC, FD, FE, FF

 

or calculate

(number of choice) x (number of choice) = 6 x 6 = 36

 

Sample of a tree diagram

 

I think what's wrong with this is that the order of the scoops doesn't matter. So where you have AB and then BA, for example, is double counting the same combination. If I understand this correctly, that's the difference between combinations and permutations (where position does or does not matter).

[DragonFaerie]

 

right you take out the repeats that give you 21

 

So they basically want the kid to write out every combination and then go through again and cross out the repeats? Is there not a shorter way to show this without getting into the formula?

[jennynd]

 

So they basically want the kid to write out every combination and then go through again and cross out the repeats? Is there not a shorter way to show this without getting into the formula?

 

 

 

for elementary, I think it is all about concept, not formula

[Arcadia]

I think what's wrong with this is that the order of the scoops doesn't matter. So where you have AB and then BA, for example, is double counting the same combination. If I understand this correctly, that's the difference between combinations and permutations (where position does or does not matter).

 

 

I have a sensory child that will scream if his ice-cream scoops are in the wrong order. His vanilla scoop should be on top of his chocolate ice-cream scoop and not vice versa :)

 

The formula way is taught in 6th grade math here. For 5th grade it is still write all out.

[DragonFaerie]

Okay. I get it. I just went back and looked at where DS had written out the problem and that's exactly what he did. The only thing was he didn't cross out all the doubles. I showed him that part (and how the problem is worded). I think we both get it now. Thank you!

[Cosmos]

 

So they basically want the kid to write out every combination and then go through again and cross out the repeats? Is there not a shorter way to show this without getting into the formula?

 

 

Yes, there is a shorter way. First, do all the ones that have A:

 

AA, AB, AC, AD, AE, AF

 

Now do all the ones that have B, but don't list any with A in them:

 

BB, BC, BD, BE, BF

 

Now do the same with C, D, E, and F:

 

CC, CD, CE, CF

DD, DE, DF

EE, EF

FF

 

That's one way to do it "methodically", i.e., ensuring no overcounting or undercounting.

[DragonFaerie]

 

Yes, there is a shorter way. First, do all the ones that have A:

 

AA, AB, AC, AD, AE, AF

 

Now do all the ones that have B, but don't list any with A in them:

 

BB, BC, BD, BE, BF

 

Now do the same with C, D, E, and F:

 

CC, CD, CE, CF

DD, DE, DF

EE, EF

FF

 

That's one way to do it "methodically", i.e., ensuring no overcounting or undercounting.

 

Great! Thank you!

[dbmamaz]

fwiw, we were just reading about this in the Murderous Maths books about Feeling Lucky or something like that. Its cool - he uses pascal's triangle as a short cut. I'm not sure my math-advanced 9 yo followed it enough to be able to do problems (murderous maths doesnt have problems), but he seemed interested

[DragonFaerie]

DS is math advanced, too, and I think that was part of my confusion. I couldn't remember him having learned much about this stuff yet and it seemed too advanced even for him. But since he's just expected to count it out, all is well.

[justasque]

I would do a ton of drawing it out, at first just in random order, then intro the concept of trees or other logical orderly methods. Next year, with that experience under his belt, I would show how the formulas work. It's critical that you *understand* how the formula is created, so that you also understand its limitation/assumptions. Experience with drawing the combos is how you get that understanding.

[daijobu]

I thought the book Competition Math for Middle School has an excellent treatment of counting for beginners. It emphasizes understanding rather than just memorizing formulas.

[DragonFaerie]

Thank you. I'll go have a look. :-)