All possible words, with our without meaning can be formed using all the letters of the word SPUTNIK. Find the probability that all vowel never occur together.

To solve this problem, we first need to find the total number of ways to arrange the letters in the word "SPUTNIK".

Step 1: Total number of arrangements The word "SPUTNIK" has 7 letters, so the total number of arrangements is 7 factorial (7!) which is 5040.

Step 2: Number of arrangements where vowels are together The vowels in "SPUTNIK" are "U" and "I". If we consider these two vowels as a single entity, we have 6 entities to arrange (SP, U, T, N, K, I). This can be done in 6! ways. However, the vowels "U" and "I" can also be arranged among themselves in 2! ways. So, the total number of arrangements where vowels are together is 6! * 2! = 1440.

Step 3: Number of arrangements where vowels are not together The number of arrangements where vowels are not together is the total number of arrangements minus the number of arrangements where vowels are together. So, it's 5040 - 1440 = 3600.

Step 4: Probability that vowels are not together The probability that vowels are not together is the number of arrangements where vowels are not together divided by the total number of arrangements. So, it's 3600 / 5040 = 0.7142857142857143 or approximately 0.71.

So, the probability that all vowels never occur together is approximately 0.71.