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License plates from different states follow different alpha-numeric formats, which dictate which characters of a plate must be letters and which must be numbers. Florida has license plates with an alpha-numeric format like the one pictured. North Dakota, on the other hand, has a different format, also pictured. Assuming all 10 digits are equally likely to appear in the numeric positions, and all 26 letters are equally likely to appear in the alpha positions, how many more license plates can Florida issue than North Dakota?

 

 

 Jul 6, 2020
 #1
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Use multiplication principle: \(26^4 \cdot 10^2 - 26^2 \cdot 10^4 = 38937600\)

 Jul 6, 2020
 #2
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Florida issues license plates in which the first three and last slots are filled with letters, and the fourth and fifth are filled with digits. Thus, there are \(26^4 \cdot 10^2 \) Florida license plates possible. North Dakota, however, issues license plates in which the first three slots are filled with letters and the last three slots are filled with digits. There are thus \(26^3 \cdot 10^3\) possible North Dakota license plates. Multiplying these out and taking the difference yields an answer of \(\boxed{28121600}\).

 Jul 6, 2020

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