Answer:
1,404,000 serial numbers can be made
Step-by-step explanation:
For the five digit serial numbers, since all letters and all numbers could be used, then we can use 26 letters (A,B,C,...Z) and 10 numbers (0,1,2,...9).
Now, say the letters and numbers are to occupy the labeled spaces below
_ _ _ _ _
1 2 3 4 5
(NOTE: Spaces 1,2,3 are to be occupied by any of the 26 letter while spaces 4 and 5 are to be occupied by any of the 10 numbers)
For space 1, there are 26 possibilities
For space 2, there will be (26 -1 ) 25 possibilities (since repetition of characters is NOT allowed)
For space 3, there will be (26-2) 24 possibilities (since repetition of characters is NOT allowed)
For space 4, we have 10 possibilities
and for space 5, we (10-1) 9 possibilities (since repetition of characters is NOT allowed)
Therefore, the number of serial numbers that can be made is
26 × 25 × 24 × 10 × 9 = 1404000
Hence, 1,404,000 serial numbers can be made
OR
Using permutation, nPr = n!/(n-r)!
For the letters,
n is the total number; n = 26
and r is the number of available spaces, r = 3
For the numbers,
n = 10
r = 2
Then, we get
26P3 × 10P2
26!/(26-3)! × 10!/(10-2)!
26!/23! × 10!/8!
15600 × 90 = 1404000
