Question:

Step 1: Identify the letters and the given condition. The word is CINEMA. Total number of letters = 6. The vowels in the word are I, E, A (3 vowels). The consonants in the word are C, N, M (3 consonants). The condition is that the order of the vowels must not change. This means that wherever the vowels are placed, their relative positions must be 'I' before 'E', and 'E' before 'A'. Step 2: Determine the number of ways to choose positions for the vowels. Since there are 6 total positions in an arrangement and 3 vowels, we need to choose 3 positions out of the 6 available positions for the vowels. The number of ways to choose these 3 positions is given by the combination formula $\binom{n}{k}$. $$ \text{Number of ways to choose positions for vowels} = \binom{6}{3} $$ $$ \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6!}{3!3!} = \frac{6 \times 5 \times 4 \times 3 \times 2 \times 1}{(3 \times 2 \times 1)(3 \times 2 \times 1)} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 $$ Once these 3 positions are chosen, the vowels 'I', 'E', 'A' must be placed in these positions in their fixed relative order (I first, then E, then A). There is only 1 way to do this. Step 3: Determine the number of ways to arrange the consonants. After choosing 3 positions for the vowels, there are $6 - 3 = 3$ remaining positions. These 3 positions must be filled by the 3 consonants (C, N, M). Since the consonants can be arranged in any order among themselves, the number of ways to arrange 3 distinct consonants in 3 distinct positions is given by $3!$. $$ \text{Number of ways to arrange consonants} = 3! $$ $$ 3! = 3 \times 2 \times 1 = 6 $$ Step 4: Calculate the total number of arrangements. To find the total number of arrangements that satisfy the condition, we multiply the number of ways to choose positions for the vowels (and place them) by the number of ways to arrange the consonants in the remaining positions. $$ \text{Total arrangements} = (\text{Ways to choose positions for vowels}) \times (\text{Ways to arrange consonants}) $$ $$ \text{Total arrangements} = 20 \times 6 = 120 $$ The final answer is $\boxed{120}$.