11 Plus Exams Forum

Can anyone please help with the following question (no answers were provided!):

I have 3 types of cuddly toy on my bedroom shelf: a teddy, a giraffe and a panda.

a. How many different ways can I arrange them on my shelf?

b. If I buy another teddy identical to the first, how many different ways can I now arrange them?


I have no idea of how to find the correct answer except by laboriously drawing the possibilites- what is the best way to approach this kind of question?

Many thanks

I always say 'if in doubt, draw it out'.
With this type of question, writing out the different combinations of t,p,g doesn't take long but must be written out systematically rather than randomly.

a) tpg, tgp, ptg, pgt, gtp, gpt = 6 ways of ordering the toys

b) ttpg, ttgp, tptg, tgtg, tpgt, pttg, ptgt, pgtt, gttp, gtpt, gtpt, gptt= 12 ways

the first part of the question falls under the heading of Factorial...

if there are 3 items, then the answer is 3 factorial, i.e. 1x2x3=6

I have 3 types of cuddly toy on my bedroom shelf: a teddy, a giraffe and a panda.

a. How many different ways can I arrange them on my shelf?

b. If I buy another teddy identical to the first, how many different ways can I now arrange them?

TGP
TPG
GTP
GPT
PTG
PGT
If you think about the three toys - any one can be put down first ie three choices. Then when that is done there are two choices for the second toy so 3 x 2 x 1

The second is more tricky - with four different toys it would be 4 x 3 x 2 x 1 but some of them look the same because the two bears are identical. If I call the teddies T you can see I get some duplicates:
TTGP
TTPG
TGTP
TGPT
TPTG
TPGT

TTGP duplicate
TTPG duplicate
GTTP
GTPT
PTTG
PTGT

TGTP duplicate
TPTG duplicate
GTTP duplicate
GPTT
PTTG duplicate
TGTP duplicate

TGPT duplicate
TPGT duplicate
GTPT dulicate
GPTT duplicate
PTGT duplicate
PGTT

ie 12 ways - the theory says (4 x 3 x 2 x 1) divided by (2 x 1) because two are identical and we can ignore these.

Thank you for such quick replies- If in doubt draw it out sounds like a good maxim for DD to remember.

Guest 55- your explanation of why it would be 3 x 2 x 1 is perfect (DD agreed). I had forgotten about factorials from all those years ago.

DD was also very impressed that I had apparently instant access to Maths whizzes!

I last did factorials when I was in school myself back in the early eighties..

When teaching this I always do the explanantion before introducing the idea that e.g. 9! is a lot easier to write than 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1