Yr 2 maths question

[petitdonkey]

DS had his maths homework tonight;

'There are 4 flavours of ice cream - vanilla, strawberry, chocolate and mint. How many different 3 scoop ice creams can you buy?'

There are then 19 images of cones with three scoops on top - ds has come up with 20 combinations that seem correct. I realise there must be a formula to work it out but I have never been much good at them - what is the correct answer???

(I'm not going to change his work I'm just interested as to how it could have been worked out)

My head is getting woozy just thinking about it.. ooh err..

Ooooh.... what a silly teacher!

I've made this mistake before...... just think - if you can't work it out in a logical way, just imagine how impossible it is to mark 25 different copies of it!!!

How many ways are there of chosing the first scoop? Then how many ways of choosing the second? Then how many for the third? Then you multiply those together to give you all the choices you have.

Thy didn't say anything about having to pick 3 different flavours, so I'd say there are 4 ways to choose scoop 1, then 4 choices for scoop 2, then 4 choices for scoop 3.

So you end up with 4 X 4 X 4 different ways of making a 3-scoop icecream

does that make any sense?

And how impossible it is for a 6yr old to work it out!! He was going at it in a totally random fashion.

Still no answer though !!

OTOH if they want you to pick 3 different flavours for each one, you would have 4 choices for scoop 1, then only 3 choices for scoop 2 (because you can't have the same again), and then only 2 choices for scoop 3.

So then you would have 4 x 3 x 2 choices of making a three-flavoured icecream.

Ahhhh - my brain is hurting too!! He has got;

VVV, VVS, VVC, VVM
SSS, SSV, SSC, SSM
MMM, MMV, MMC, MMS
CCC, CCM, CCS, CCV
CMS, CMV, MSC, VSC

So that would be 24??? (OMG, I used to teach Yr 6 )

er.... isn't that 20?

Lovecheese - sorry, that was at the 4x3x2!! I did count DSs to be 20

Isn't it 4! (= 4 x 3 x 2 x 1= 24) assuming you can't pick a flavour more than once.

Eh??? stupid

Sorry!! That read as rather terse! I meant I'm being stupid, not you at all!!

my word,I think I'm struggling with this,nevermind a yr 2!!!

I think the question needs to be worded more clearly,tbh.

Well DUH!

Let us say there are five flavors of icecream: banana, chocolate, lemon, strawberry and vanilla. You can have three scoops. How many variations will there be?

Let's use letters for the flavors: {b, c, l, s, v}. Example selections would be

{c, c, c} (3 scoops of chocolate)
{b, l, v} (one each of banana, lemon and vanilla)
{b, v, v} (one of banana, two of vanilla)
(And just to be clear: There are n=5 things to choose from, and you choose r=3 of them.
Order does not matter, and you can repeat!)

Now, I can't describe directly to you how to calculate this, but I can show you a special technique that lets you work it out.

Think about the ice cream being in boxes, you could say "move past the first box, then take 3 scoops, then move along 3 more boxes to the end" and you will have 3 scoops of chocolate!

So, it is like you are ordering a robot to get your ice cream, but it doesn't change anything, you still get what you want.

Now you could write this down as (arrow means move, circle means scoop).

In fact the three examples above would be written like this:

{c, c, c} (3 scoops of chocolate):
{b, l, v} (one each of banana, lemon and vanilla):
{b, v, v} (one of banana, two of vanilla):
OK, so instead of worrying about different flavors, we have a simpler problem to solve: "how many different ways can you arrange arrows and circles"

Notice that there are always 3 circles (3 scoops of ice cream) and 4 arrows (you need to move 4 times to go from the 1st to 5th container).

So (being general here) there are r + (n-1) positions, and we want to choose r of them to have circles.

This is like saying "we have r + (n-1) pool balls and want to choose r of them". In other words it is now like the pool balls problem, but with slightly changed numbers. And you would write it like this:

where n is the number of things to choose from, and you choose r of them
(Repetition allowed, order doesn't matter)
Interestingly, we could have looked at the arrows instead of the circles, and we would have then been saying "we have r + (n-1) positions and want to choose (n-1) of them to have arrows", and the answer would be the same ...

So, what about our example, what is the answer?

(5+3-1)! = 7! = 5040 = 35

3!(5-1)! 3!×4! 6×24

I think we're all clear now

I need a drink reading all this. Be still my spinning head...

Febreeze!! Please tell me you Googled that???

So, our current answers are 20, 24.... Any more? AND I still don't know why there were 19 cones printed on the sheet!!
And I still don't know what the formula would be. (Why am I bothering?? DS is asleep and I'm going to hand it in tomorrow anyway....)

Hell Yes

DH is my maths geek but he's currently in Las VEgas playing the World Series of Poker with other maths geeks.

I think your DS (in your 19:38:02 post) is exactly right, petitdonkey, and his [assuming it was his not yours] writing out was systematic too - well done him! Here's how I'd think about it:

• no right thinking person cares about the order in which scoops are arranged in their cone; they only care about how many scoops they have of what kinds
• we could choose all three scoops the same, all three scoops different, or two scoops of one with one scoop of another
• there are four choices that have three scoops the same (just choose which flavour it's going to be)
• there are four choices that have all three scoops different (because you only have to choose which flavour to leave out)
• to see how many choices there are for the two of one, one of another kind of ice cream, first decide which kind you'd like to have two of, and then choose a different kind for your other scoop. For each of the four flavours you could have two of, there'll be three flavours you could have the third scoop be (remembering that it has to be different). That's 12 choices.

Altogether there are 4 + 4 + 12 = 20 different icecreams.

I'm sure in Y2 he's not expected to do calculations involving factorials. Systematic counting is perfect.

Sidheag - You sound like you know what you're talking about!! He didn't work it out quite that systamatically but that is the combination he came up with - just gave him a nudge as to how he could make sense of what he had done so far IYSWIM.

I didn't think that he needed to work out a formula, I was just annoyed with myself that I couldn't work out quickly what the correct answer is!! It also really threw me that there were only 19 printed cones (and no space to add more).

Thank you everyone - it's in his bag and will be out of my house at 8am!!

Systematically - and English was always my strong point....

I came up with 24 (there is a formula for working this out but I have forgotten it and I bet most primary teachers wouldn't know it let alone a year 2 pupil - so I did it logically):

3 the same flavour - 4 different options for this = 4
2 the same and one different - 4 x 3 options for this = 12
3 different flavours - 4 x 2 options for this = 8

no idea why there would be 19 images of cones ?

I think for year 2 level they are just looking for evidence of understanding there are a number of choices. Easiest way is to get 4 groups of things and label as each type, and then pick out combinations of 3 things and note them down.

oops - 3 different flavours - my method would come up with the same option twice - so need to lose 4 for that - would bring it down to 20.

This is a bloody daft question!

The teacher needs to specify:

1. does it matter which order the flavours are stacked - should the children distinguish between a vanilla-chocolate-strawberry cone and a vanilla-strawberry-chocolate?

1. are you allowed to use each flavour more than once.

If you're not allowed to repeat flavours, and it doesn't matter which way the ice creams are ordered the answer is 4 Choose 3 (4!/(3!*1!))=4: you can work this out by choosing a flavour to miss each time - there are 4 choices here.

If you're not allowed to repeat flavours and it does matter how the ice creams are ordered, the answer is 4!/1! = 24. Four choices for bottom scoop, 3 for middle scoop, 2 for top scoop.

If you're allowed to repeat flavours and it doesn't matter which way the ice creams are ordered, there are:
4 ways of getting 3-flavour cones (VSC, VSM, VCM, SCM)
12 ways of getting 2-flavour cones (VVS, VVC, VVM, VSS, VCC, VMM, SSC, SSM, SCC, SMM, CCM, CMM)
4 ways of getting 1-flavour cones (VVV, SSS, MMM, CCC)
So 20 choices in all.

If you're allowed to repeat flavours and it does matter which way the ice creams are ordered then you have 444 choices - i.e. 64.

What a question to give to Y2 children! Sadly, many teachers of primary school children lack sufficient mathematical knowledge and ability to get them through the primary curriculum successfully.

Is this not factorials which is A-level maths. And you use the factorial button on the calculator.