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# Permutations watch

1. Consider the words TWENTYONE, what is the probability that the letters will be drawn such that the word ONE is spelt out?

What I did was (7 x 6!) / 2!

7 spaces where O can come in and there are two Es so its divided by 2! hence probability that ONE is spelt out = 1 / 18

Anything wrong with that?
2. No I'm afraid that's wrong, why have you used 7 x factorial (!) 6 in your first expression? Also you do not divide by 2.

Think agian, it's quite simple
4. I think I follow the reasoning: You have to have "ONE" together, so the O can be on any of the 7 first places, and then N must come after O and there is only one N so no problems with that (just regard "ON" as one letter really). There are two E's, so either of those E's can be in the place after "ON" and thus you need to divide by 2! How the other 6 letters (including the other E) can be arranged is 6! and that won't affect the "ONE", so just multiply by it.

edit: I realise this probably wasn't much clearer...
5. So is that acceptable? How did u go about it then DFranklin?
6. Nota: There are two 'N's in TWENTYONE

What is also bothering me about both your solutions is that at no point do you actually explain what you are dividing by.

As to how I did it: I cheated and did a brute force search by computer. (Which is what I usually do for these kinds of problems when posted - the computer check is very very reliable).

But by hand, I would consider TWENTYONE as , so we have 9 distinct symbols.

The O can go in any of 7 places. Either N1 or N2 can follow, and then either E1 or E2 can follow that. There are 6! ways of placing the other 6 symbols. So there are 7x6!x2x2 = 4x7! ways of spelling ONE.

There are 9! total permutations. So p(can spell ONE) = 4x7! / 9! = 4/(8x9) = 1/18.
7. Right thanks. But I think I'll use my way first and see what happens.. if it turns out to be wrong I'll make sure I won't repeat this method again!
8. Just be aware, I wouldn't be happy to give more than about half marks for what you've posted. Depending on mark scheme, I might not give more than 1 mark. Explanations matter.

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