We are asked to find the probability that if we randomly select the given letters, they spell out GEOMETRY. First, let's find the number of distinct permutations of the letters.
The number of distinguishable permutations of n objects in which one of the objects is repeated r_1 times, the other r_2 times, and so on, can be calculated using the following formula.
n!/r_1 ! * r_2! * ... * r_k !
In total there are 8 letters and the letter E is repeated 2 times. No other letter is repeated. Let's substitute these values into the formula.
We found that the total number of distinct permutations of the letters is 20 160. Since there is only one distinct permutation of the letters that spells out GEOMETRY, the probability that a random permutation of the letters would spell out this word is 120 160.
