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I am given the following question:

A code consists of $4$ letters from the English alphabet and $3$ letters from the Greek alphabet. There are $8$ English letters and $6$ Greek letters to choose from and repetition is not allowed. How many $7$-letter codes are possible?

I tried solving this question by finding the number of possible combinations for the English Letters ($8C4$) and Greek Letters ($6C3$). I multiplied the two to get $1400$ different $7$ letter combinations. I multiplied $1400$ by $7!$ since there can be $7$ possible letters in the first slot, followed by $6$, and so on.

The answer that was given was $201600$ and my teacher stated that this was the case because it should be $(8P4) \times (6P3)$. Please explain what I did wrong.

Thank You :)

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    $\begingroup$ Your teacher is mistaken because his/her method is not looking at the relative order in which the English and Greek letters can be arranged with respect to each other. $\endgroup$ – Anurag A May 1 '18 at 23:09
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    $\begingroup$ Your teacher would be correct if the English letters had to precede the Greek letters. $\endgroup$ – N. F. Taussig May 1 '18 at 23:10
  • $\begingroup$ Is there anything wrong with my method? $\endgroup$ – John Yu May 1 '18 at 23:31
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    $\begingroup$ Your solution is correct. $\endgroup$ – N. F. Taussig May 1 '18 at 23:52
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Each method is solving this under different conditions, and getting different results.

Your teacher's method is simply finding the ways you can arrange the English letters into groups of 4, then the same for the Greek letters in groups of 3, and then puts them together. In this way, your teacher's method assumes that the code is of the form $\Gamma\Gamma\Gamma EEEE$ OR $EEEE\Gamma\Gamma\Gamma$ ($\Gamma$=Greek letter, $E$ =English letter)

Your method can arrange them in any way, and I believe that is likely to be the intention of the question.

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