In how many different ways can the letters of the word ‘TOMATO’ be arranged?

Concept: When n objects are arranged in a row and some objects repeat, the number of distinct arrangements is n! divided by the factorial of each repeated group's count. This correction removes arrangements that look identical because the repeated letters are indistinguishable from each other.

Application: The word ‘TOMATO’ has 6 letters: T, O, M, A, T, O.

1. Total letters n = 6, so without any repetition there would be 6! = 720 arrangements.

2. The letter T occurs 2 times and the letter O occurs 2 times; M and A occur once each.

3. Divide by 2! for the repeated T’s and 2! for the repeated O’s: 720 / (2! × 2!) = 720/4.

4. This gives 180 distinct arrangements.

Cross-check: Choose 2 of the 6 positions for T in C(6,2) = 15 ways, then 2 of the remaining 4 positions for O in C(4,2) = 6 ways, then arrange the remaining letters M and A in the last 2 positions in 2! = 2 ways: 15 × 6 × 2 = 180 — the same result confirms the count.

Hence, the number of different ways to arrange the letters of ‘TOMATO’ is 180.