Questions: Serial numbers for a product are to be made using 2 letters followed by 2 digits. The letters are to be taken from the first 8 letters of the alphabet, with no repeats. The digits are taken from the 10 digits (0,1,2, ..., 9), with no repeats. How many serial numbers can be generated?

Serial numbers for a product are to be made using 2 letters followed by 2 digits. The letters are to be taken from the first 8 letters of the alphabet, with no repeats. The digits are taken from the 10 digits (0,1,2, ..., 9), with no repeats. How many serial numbers can be generated?

Transcript text: Serial numbers for a product are to be made using 2 letters followed by 2 digits. The letters are to be taken from the first 8 letters of the alphabet, with no repeats. The digits are taken from the 10 digits $(0,1,2, \ldots, 9)$, with no repeats. How many serial numbers can be generated? $\square$

Solution

Solution Steps

Step 1: Calculate the number of ways to select letters

Using the permutation formula $P(n, r) = \frac{n!}{(n-r)!}$, where $n = 8$ and $r = 2$, the number of ways to select 2 letters from the first 8 letters of the alphabet without repetition is: $P(8, 2) = \frac{8!}{(8-2)!} = 56$.

Step 2: Calculate the number of ways to select digits

Similarly, using the permutation formula for digits where $n = 10$ and $r = 2$, the number of ways to select 2 digits from 0 to 9 without repetition is: $P(10, 2) = \frac{10!}{(10-2)!} = 90$.

Step 3: Calculate the total number of unique serial numbers

By multiplying the number of ways to select letters by the number of ways to select digits, the total number of unique serial numbers that can be generated is: $56 \times 90 = 5040$.