How many anagrams with a specific subword?

Question

I'm trying to find the number of anagrams of "ACKNOWLEDGEMENT" (15 characters) that contain the subword "EDGE".
I know the total number of anagrams is $frac{15!}{3! cdot 2!}$. I'm not sure of what to do after this. I know this is an overcount.

Joffan · Accepted Answer

Hopefully you understand that the number of anagrams you have correctly given has the division by $3!cdot 2!$ because of the $3$ E's and $2$ N's in the available letter pool.
To get the number of full anagrams containing "EDGE", you can simply regard $fbox{EDGE}$ as a single "letter" that you anagram with the other $11$ items in the letter pool. Then you have $12$ items to permute with only the $2$ copies of N to adjust for, using the same calculation as you already exhibited.

Adam Rubinson · Answer

The number of ways EDGE can be the first $4$ letters is $1$. The number of ways the remaining letters can be arranged is then $frac{11!}{2!}.$
The number of ways EDGE can be the $2nd$ to $5th$ letters is $1$. The number of ways the remaining letters can be arranged is then $frac{11!}{2!}.$
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So the answer is $frac{11!}{2!} times 12.$

How many anagrams with a specific subword?

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