How many arrangement can be made from the word COMMERCE, such that all the vowels do not come together ?

To solve this problem, we first need to find the total number of arrangements of the word COMMERCE and then subtract the number of arrangements where all the vowels come together.

Step 1: Total Arrangements of the word COMMERCE The word COMMERCE has 8 letters, but the letters M and C are repeated twice. So, the total number of arrangements is 8!/(2!*2!) = 20160.

Step 2: Arrangements where all vowels come together The vowels in the word COMMERCE are O, E, E. If we consider these vowels as a single entity, then we have 6 entities (C, M, M, C, R, and the entity OEE). These 6 entities can be arranged in 6!/(2!*2!) = 360 ways. But the vowels O, E, E can be arranged among themselves in 3!/2! = 3 ways.

So, the total number of arrangements where all vowels come together is 360*3 = 1080.

Step 3: Arrangements where all vowels do not come together This is simply the total number of arrangements minus the number of arrangements where all vowels come together. So, the answer is 20160 - 1080 = 19080.

Therefore, there are 19080 arrangements of the word COMMERCE such that all the vowels do not come together.