SOLUTION: a) FIND NUMBER OF DIFFERENT CODES CONSISTING OF 3 LETTERS FOLLOWED BY 4 DIGITS THAT CAN BE OBTAINED FROM A,B,C,D 1,2,3,4,5,6? b)FIND THE NUMBER OF 3-LETTER CODE WORDS THAT CA

If you cannot repeat letters or digits:

Choose the first letter 4 ways.
Choose the second letter 3 ways.
Choose the third letter 2 ways.
Choose the first digit 6 ways.
Choose the second digit 5 ways.
Choose the third digit 4 ways.
Choose the fourth digit 3 ways.

That's 4×3×2×6×5×4×3 = 8640

If you CAN repeat letters or digits:

Choose the first letter 4 ways.
Choose the second letter 4 ways.
Choose the third letter 4 ways.
Choose the first digit 6 ways.
Choose the second digit 6 ways.
Choose the third digit 6 ways.
Choose the fourth digit 6 ways.

That's 4×4×4×6×6×6×6 = 82944

That can be interpreted as

1. The two P's MUST be used

or

2. The two P's MAY be used.
 
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1. If the two P's MUST be used, there are 3 forms:

_PP, P_P, and PP_

And we can fill the blank 4 ways with A, L, E or S

That's 3×4 = 12 ways

2. If the two P's MAY be used, but don't have to be, 
then in addition to those 12 ways, we can choose 3
letters from {A,P,L,E,S}

Choose the first letter 5 ways.
Choose the second letter 4 ways.
Choose the third letter 3 ways.

12 + 5×4×3 = 12 + 60 = 72 ways.


Edwin