# How many anagrams with a specific subword?

Mathematics Asked by hhhiuw on February 24, 2021

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.

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.

Correct answer by Joffan on February 24, 2021

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 $$frac{11!}{2!}.$$

...

So the answer is $$frac{11!}{2!} times 12.$$