**Quote:**

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.