What is the probability that an arrangement of the letters of the word…

Concept

When a word's letters include repeats, the count of distinct arrangements is not simply n! — repeated letters are divided out via their factorial. The probability that a random arrangement satisfies a condition (like starting with a particular letter) is the ratio of favourable arrangements to total distinct arrangements: P = (favourable arrangements) / (total distinct arrangements).

Application

1. STUDENT has 7 letters: S, T, U, D, E, N, T — the letter T occurs twice.

2. Total distinct arrangements of all 7 letters = 7! / 2! = 5040 / 2 = 2520 (dividing by 2! for the repeated T).

3. Arrangements starting with T: fix one T in the first position; the remaining 6 letters (S, U, D, E, N, T) now contain only one T, so they arrange in 6! = 720 ways.

4. Probability = favourable / total = 720 / 2520 = 2/7.

Cross-check

Independently: in a uniformly random distinct arrangement, each letter instance is equally likely to occupy the first position. STUDENT has 2 T's out of 7 total letters, so P(first letter is T) = 2/7 — the same answer as the counting method above.