# Distinct permutations of the word “toffee”

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

-
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

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!

-
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

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.

-