Combinations and permutations

[starlingtree]

In which year do they learn this topic? Thanks.

Eg 1

A group of 3 whole numbers which may or may not be different are such that their highest common factor is 10 and their lowest common multiple is 100. How many such groups are possible?

Eg 2

Palindromes are numbers that read the same forward and backwards. Find the number of palindromes that can be formed using all the digits 3, 5, 8 and 0. Each digit can be repeated twice at maximum.

Year 5/6

Can you tell me the answers to these questions?

1. The highest common factor is 10, so all the numbers are multiples of 10 e.g. 10,20,30,40 ... The lowest common multiple is 100.

100=2x2x5x5 so every number must consist of some combination of these factors
The only possibles are 10,20,50,100

You must have either the 20 and the 50 or the 100 (or there would be a lower LCM)
so combinations are (satisfying HCF=10)
10,20,50
20,20,50
20,50,50
10,10,100
10,20,100
10,50,100,
20,50,100

2.palindromes with 0,3,5,8 using every number and each number at most twice

First consider 8 digit palindromes using each number twice e.g. 35800853
there are 24 ways of arranging 4 numbers (4x3x2x1). however we don't want any of the numbers starting with 0 (I assume) so that makes it 18 ways (3x3x2x1)

We also need to consider digit palindromes e.g. 3580853 (so the number in the middle is used once; the others twice).
This also comes down to the number of ways you can arrange 4 numbers without starting with 0 - so 18 again.

so that's 36 permutations in all.

I assume these are stretch questions? The maths isn't hard, but they need a bit of lateral thinking.

I don’t think q1 is permutations and combinations. I like@redskydelight ‘s solution.

q2, in my opinion, is not stated clearly. Does “each digit can be repeated twice at maximum” mean can occur at most twice, or can be seen next to itself in the number at most twice, so for example 335808533 is OK but 33358085333 isn’t?

Can the palindrome start with zero?

I think the questions in the OP are interesting and challenging.

My previous post was written without the benefit of reading @redskydelight ‘s answer to q2.
Within its assumptions, I agree with it and admire its clarity.

Definitely stretch questions if given to KS2 or 3, I would have thought.

(Not a teacher)

I haven't come across anything like either example up to GCSE.
The closest would be to find one example that meets the criteria.

I would go further to assert that for combinations & permutations you usually have to know the factorial ! symbol and I don't believe ! is on the GCSE syllabus.
However I could have missed it or it could be in the higher numbered grades my DC haven't accessed. @noblegiraffe will set me right no doubt.

That isn't to say it couldn't be an interesting off piste stretch piece of homework for higher sets where the method they use is more interesting than getting the correct answer.

I don't think questions like that would be given to manny classes as a whole.
More likely to be given as a stretch question to someone who had finished work early.

There is nothing particularly difficult about the maths. It is the logic required to get to an elegant solution like given above that is the tricky bit. It could be done, with a lot of effort and probably incompletely, by primary kids.

Thank you all, and thanks @redskydelight for your detailed explanation.

My Y8 son brought this home from school and asked me for some help, he says the teacher gave it to him as a nice challenge, but I didn't know where to start!

Perms and Combs as a proper topic has now been moved to Further Maths A-level (used to be on A-level). What has been added to GCSE recently is the 'Product Rule for Counting' which deals with stuff like the number of possible meal combinations ordered from a short menu or ways of arranging some letters or numbers. Nothing as difficult as in the OP though.

www.mathsgenie.co.uk/resources/6-product-rule-ws.pdf

Thanks so much for this. Much appreciated.

Wow. I'm surprised P&Cs is now a FM topic. I have no idea at what stage I did them but it seems instinctively 'easier' than FM. But I guess you can't do everything earlier.

A bit like bases. I'm pretty certain I did converting to/from binary before y9, and that I don't think is on the maths GCSE syllabus (though I wouldn't be surprised if it were on Computer Science GCSE).

it seems instinctively 'easier' than FM

Kids can't do them! Even FM kids struggle!

I used to love teaching them as part of the S1 module but I'd say it took me a good 2-3 years of teaching to be confident I had my head round it and could do questions without pre-prep.

Oh fair enough. Having forgotten all of my uni maths and most of my A level stuff P&Cs still seems quite fresh in my mind. Maybe it is from doing puzzles or something (or maybe if I had to do some questions I'd find I have forgotten more than I realise.)

You mean "even FM teachers struggle"

This is why it is good to start doing something like OP's questions much earlier.