Original Question:
How many different strings can be made from the letters in "EVERGREEN"
using at least 7 of it's letters.
Note that the 4 "E"s and 2 "R"s are indistinguishable.
I understand how to use permutations with repetition to determine the number of ways for 9 total letters, but the "at least 7 letters" part stumps me.
Without that condition, it would be 9!/4!2! possible permutations.
Now, should I simply subtract some other number of ways? Perhaps the number of permutations that are less than 7?
How should I proceed? Thanks!
Best Answer
using 9 letters the number of different words is $\binom{9!}{4!2!}$
to find how many words you can form with 8 letters you have to perform the same calculation on each of the following set of letters
EEEVRRGN, EEEERRGN, EEEEVRGN, EEEEVRRN, EEEEVRRG
e.g. for the first one the number of words is $\binom{8!}{3!2!}$, and so on
then you have to add to the previous results the same calculation for 7 letters considering all the different sets, and precisely
EEVRRGN, EEERRGN, EEEVRGN, EEEVRRN, EEEVRRG, EEEERGN,EEEERRN,EEEERRG, EEEEVGN, EEEEVRN,EEEEVRG, EEEEVRR
I don't know whether there exists a different direct way