how many ways can all the letters in the word..be arranged

[wind]

in how many ways can all the letters in the word CANADA be arranged if the consonance must always be in the order that they occur in the word itself?
6!/3!

Possible ways of arranging CND
3!=6
cdn- wrong order
dnc- worng
ncd-wrong
ndc-wrong
cnd-right
dcn-wrong

...now what do I do

Thanks

 

[soroban]

Re: how many ways can all the letters in the word..be arrang

Hello, wind

in how many ways can all the letters in the word CANADA be arranged
if the consonants must always be in the order that they occur in the word itself?

The consonants \(\displaystyle C,\,N,\,D\) must be in that order.
. . (Why are you re-arranging them?)

Then then are four "slots" in which to place the A's: .
C
N
D


If the three \(\displaystyle A\)'s are together, there are 4 ways.
If two \(\displaystyle A\)'s are together and the third is apart, there are: \(\displaystyle 4\cdot3\,=\,\)12 ways.
If the three \(\displaystyle A\)'s are separated, there are 4 ways.

Therefore, there are: \(\displaystyle \,4\,+\,12\,+\,4\:=\:\fbox{20}\) ways.

 

[wind]

thanks soroban

 

[pka]

There is a very easy way to see this problem.
The letters {C,N,D} can be arranged in (3!)=6 ways.
Thus in (1/6) of the total will the letters be in that particular order.