Discussion
Explore all 5 letter combinations formed using the letters a through h. How many of these words contains no repeats, also do not include the subword "bad" ?
Answer: A
There are 6660 such words.
Words of length 5 with no repeated letter from 8 letters:
\[ P(8, 5) = 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 = 6720 \]
Words containing 'bad': treat 'bad' as one block, then choose 2 more distinct letters from the remaining 5 and arrange the block with them.
\[ \begin{aligned} \binom{5}{2} \times 3! &= 10 \times 6 = 60 \\ 6720 - 60 &= 6660 \end{aligned} \]
The other figures do not follow from this count: subtracting the 60 forbidden words is the only correction needed, since 'bad' cannot occur twice in five distinct letters.
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