Permutations of 9 total letters, using at least 7 letters. 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!

If you use exponential generating functions you can reduce the problem to computing the coefficient of $x^7/7!$ in the expansion of $(1+x)^3(1+x+x^2/2!)(1+x+x^2/2!+x^3/3!+x^4/4!)$.
This is messy but not overwhelming to do by hand if one uses appropriate tricks (eg replace $(1+x+x^2/2!+x^3/3!+x^4/4!)$ with $(x^3/3!+x^4/4!)$ and $(1+x)^3$ with $(3x+3x^2+x^3)$ since the other terms don't contribute to the $x^7$ term)
The most reasonable alternative is to divide up the counting into cases based on the possible partitions that may result from such a set:


*

*Taking 4 E's and 1 each of three out of V,R,G, and N.

*Taking 4 E's, 2 R's, and 1 each of two out of V, G, and N.

*Taking 3 E's, and 1 of each of V,R,G, and N.

*Taking 3 E's, 2 R's, and 1 of each of V,G, and N.

*Taking 2 E's, 2 R's, and 1 of each of V,G, and N.

