What is the number of distinguishable permutations of the letters in the word oregon?

360.

There are 6 letters, so there are 6! (=720) different permutations of 6 letters.

However, since the two 'o's are indistinguishable, it is necessary to divide the total number of permutations by the number of permutations of the letter 'o's - 2! = 2

Thus 6! ÷ 2! = 360