Option 4 : 1680

**Given:**

Word = 'NOMINATION'

**Formula Used:**

n! = n(n - 1)(n - 2)(n - 3)......(3)(2)(1)

**Calculation:**

Total number of letters in the word = 10

Position of A and T are fixed.

So, A _ _ _ _ _ _ _ _ T

Remaining place = 10 - 2 = 8

Total number of ways of writing left most place = 1 way

Total number of ways of writing right most place = 1 way

Total number of ways of writing remaining 8 places = 8! ways

Letters I and O both are repeated two times, So divide by 2! and 2!

Letter N is repeated three times, So divide by 3!

The required total number of ways = (1 × 8! × 1)/(3! × 2! × 2!)

⇒ (8 × 7 × 6 × 5 × 4 × 3!)/(3! × 2! × 2!)

⇒ 8 × 7 × 6 × 5

⇒ 1680

**∴ The possible total number of different ways will be 1680.**