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What does distinct permutations mean and how many distinct permutations can be formed from all the letters of word TOFFEE?

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Distinct permutations in this case means distinct $6$-letter "words" that can be made using each letter once and only once. TOFFEE is such a word, as is FEETOF. –  André Nicolas Jul 26 '12 at 21:46
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3 Answers

We know that the number of permutations of some given string of length $n$ is $n!$, however, we need to take into account the number of repeated permutations, we do this by counting the number of permutations of the repeated letters (in this case $F$ and $E$).

Therefore, we have:

$$\frac{6!}{2!^{2}}=180$$

Hope this helps!

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i thought Distinct permutations mean that exclude repetitions. is it wrong? –  user1419170 Jul 26 '12 at 21:26
    
@user1419170 Yes, that is why we are dividing by the repeated permutations (of which there are $2!\times2!=2!^2$). –  Shaktal Jul 26 '12 at 21:26
    
ok thanks now clear –  user1419170 Jul 26 '12 at 21:33
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Here is another way to think about it. You have the word TOFFEE and six blanks

- - - - - -

you want to place the letters in. Begin with the E. Choose two of the six blanks and pop the Es in. This can be done ${6\choose 2}$ ways. Now four blanks remain; place the Fs in these. There are ${4\choose 2}$ ways to do this. Two blanks remain for T and O; there are two ways to do this. So you get in toto, $$2{6\choose 2}{4\choose 2} = 2\cdot 15 \cdot 6 =180$$ ways to permute TOFFEE.

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The term "distinct permutations" takes into account that the word TOFFEE has two F's and two E's. This means that if we simply swap the two F's that the permutation is considered the same. You have to take this into account when doing the calculations for this problem.

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