a. In how many ways can the letters in the word CARLETON be arranged so that it contains either CA or AC as sub-words?

a. In how many ways can the letters in the word CARLETON be arranged so that it contains either CA or AC as sub-words?
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Jayden-James Duffy

The word CARLETON have 8 letters. We want to arranged this word so that it contains either CA of AC as sub-words.
The letters C and A can be grouped and considered as a single letter. that is, (CA)RLETON.
Hence we can assume total letters as $1+6=7$ and all are different. Number of ways to arrange these letters is $7!=5040$. Now, CA or AC can be arranged themselves in $2!=2ways$, Therefore, the total number of arrangements is $5040×2=10080$
Thus, the total number of arranged so that the word contains either CA or AC as sub-words are 10080.