how many different string can be made from the word PEPPERCORN when all letters are used and such strings do not contain the substring CON?

Question

anonymous · Answer

you can rearrange PEPPERCORN  in 10! ways. then, we can consider  CON as one letter, lets say  Z , so instead of PEPPERCORN, we'll have  PEPPERRZ, we can rearrange this in 8! ways. so I think the answer is 10!-8!

kainui · Answer

Almost right, but not quite.

See, peppercorn has 10 letters in it, so it is 10!, however you have to divide out all the ways in which you can rearrange the letters that are being reused, such as P, E, and R.

So P appears 3 times, E twice, and R twice so
$$\frac{ 10! }{ 3!*2!*2! }$$

That tells us how many different arrangements you can have, but now we need to get rid of all the occurrences of "CON".

Simply enough, we can just look at the ten letter word and see by counting (or more complicated factorial work) that you can put CON in only 8 possible places, so just subtract 8 from the factorial answer above:

$$\frac{ 10! }{ 3!*2!*2! }-8$$

I believe that's the right answer, but I'm human too!

kainui · Answer

Wait, subtracting 8 was ridiculous lol. It too is a factorial that must be calculated similarly I don't know what I was thinking.

raden · Answer

the possibles for word "PEPPERRZ" is 8!/(3!*2!*2!), @Kainui

kainui · Answer

Wait, that's not the final answer though.

kainui · Answer

So that tells you all the occurrences that will contain CON in it, while the other calculation I did will tell how many total words you can make.

Seems we can just subtract the two and get the final answer:

$$\frac{ 10! }{ 3!*2!*2! }-\frac{ 8! }{ 3!*2!*2! }$$

raden · Answer

yeah, i meant the finally it will be :
10!/(3!*2!*2!) - 8!/(3!*2!*2!)