Question

I'm trying to build a 12-letter word with 4 X's, 4 Y's, and 4 Z's. In how many ways can I build a word if there are no X's in the first 4 letters, no Y's in the next 4 letters, and no Z's in the last 4 letters?

Answer 1

@satellite73

Answer 2

so i need to ask satellite73?

Answer 3

Well he is the Legend.

Answer 4

but you can only send him messages if your his fan

Answer 5

i'm his fan but i can't send messages too him

Answer 6

i have no idea how to do this, but we could at least try something

Answer 7

well, here is the Legend.

Answer 8

cool

Answer 9

but how

Answer 10

you have four slots in which to put only y and z

Answer 11

ok

Answer 12

for some reason i can't send you a message even though i'm your fan

Answer 13

just off the top of my head, how many ways can this be done?
1) all y: 1 way
2) 1 z 3 y: 4 ways
3) 2z 2y: 6 ways
4) 3z 1 y : 4 ways
5) all z : 1 way
so far we have \(16=2^4\) possibilities

Answer 14

so its 16?

Answer 15

oh hell no, we have lots more work to do

Answer 16

can you fan me so i can send you questions

Answer 17

what about the next 4 slots ? it can have no y, only z and x so i guess we have to work on what happened above

Answer 18

ok

Answer 19

if
5) we used up all the z there is only 1 way (all x)
4) used up 3 z there are 1z, 3x : 4 ways
3) used up 2 z : 2z, 2x : 6 ways
2) used up 1 z : 3z, 1z, 4 ways
1) used up no z : 4z 1 way

Answer 20

there must be an easier way to do this....

Answer 21

i think so too

Answer 22

well maybe not
a tree diagram might help

Answer 23

but how

Answer 24

Is this right:
The first 4 letters will be y and z. There are 16 arrangements of those 4 letters.
The next group of 4 will be X and Z and there will be 16 arrangements.
The last group of 4 will be X and Y and there will be 16 arrangements. So??? 16^3

Answer 25

so the answer is 16^3?

Answer 26

but its wrong

Answer 27

i tried it already

Answer 28

yeah i think there is something else going on here

Answer 29

if you make a tree diagram of sorts you might see it, then add up the possibilities
use pencil and paper

Answer 30

i tried that but i didn't get anything

Answer 31

Well then it might be that some of the arrangements look the same because the letters are repeated .

Answer 32

then what can you get

Answer 33

Did you try 2^12 since there are 2 choices for each position?

Answer 34

yeah
and it didn't work

Answer 35

got it
its 346 ways