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# help counting

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How many 3-letter words can we make from the letters A, B, C, and D, if we are allowed to repeat letters, and we must use the letter A at least once, and the letter B at least once?

Aug 5, 2022

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We already know 2 of the digits, so there are 4 choices for the final digit.

Now, we have to order them. Divide this into 2 cases.

Case 1: All 3 digits are distinct.

There are $$3! = 6$$ ways to do this, and there are 2 options (C or D), which makes for $$6 \times 2 = 12$$ options.

Case 2: The 3rd digit is A or B

There are $$3! \div 2! = 3$$ ways to order them, and there are 2 options (A or B), which makes for $$3 \times 2 = 6$$ options.

So, there are $$12 + 6 = \color{brown}\boxed{18}$$ words.

Aug 5, 2022