3-letter codes:

Choose the first card 5 ways.
Choose the second card any of the remaining 4 ways.
Choose the third card any of the remaining 3 ways.

That's 5*4*3 = 60

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4-letter codes:

Choose the first card 5 ways.
Choose the second card any of the remaining 4 ways.
Choose the third card any of the remaining 3 ways.
Choose the fourth card either of the remaining 2 ways.

That's 5*4*3*2 = 120

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5-letter codes:

Choose the first card 5 ways.
Choose the second card any of the remaining 4 ways.
Choose the third card any of the remaining 3 ways.
Choose the fourth card either of the remaining 2 ways.
Choose the fifth card that 1 remaining way.

That's 5*4*3*2*1 = 120

[Notice that the number of 5-letter codes is the same
as the number of 4-letter codes, because for every 4-letter
code, just tack on the remaining card and you have a 5-letter
code.]

Edwin