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License plates consist of sequence of 3 letters followed by 3 digits. How can they be arranged if

(i) no repetition of letters is permitted, how many possible license plates are there?

should it be $26P3 \cdot (10^3)$ or $3! \cdot (10^3)$?

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Your first answer is correct. Since repetition of letters is not permitted, there are $26$ ways of choosing the first letter, $25$ ways of choosing the second letter, and $24$ ways of choosing the third letter. Therefore, there are $P(26, 3) = 26 \cdot 25 \cdot 24$ ways of selecting the three letters. Since repetition of digits is permitted, there are ten choices for each of the three digits. Thus, the number of license plates that can be formed is $$P(26, 3) \cdot 10^3 = 26 \cdot 25 \cdot 24 \cdot 10^3 = 15,600,000$$

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