Introduction:
In this problem, we are required to calculate the probability of not getting any vowel when 2 letters are taken at random from the word HOME.

Solution:
To solve this problem, we need to determine the total number of ways in which 2 letters can be selected from the word HOME, and then we need to determine the number of ways in which we can select 2 consonants from the word HOME. We can then calculate the probability of selecting 2 consonants as follows:

Step 1: Determine the total number of ways in which 2 letters can be selected from the word HOME.
There are 4 letters in the word HOME. Therefore, the total number of ways in which 2 letters can be selected from the word HOME is given by:

4C2 = (4!)/(2!*(4-2)!) = 6

Therefore, there are 6 ways in which 2 letters can be selected from the word HOME.

Step 2: Determine the number of ways in which we can select 2 consonants from the word HOME.
The word HOME has 2 vowels (O and E) and 2 consonants (H and M). Therefore, the number of ways in which we can select 2 consonants from the word HOME is given by:

2C2 = 1

Therefore, there is only 1 way in which we can select 2 consonants from the word HOME.

Step 3: Calculate the probability of selecting 2 consonants.
The probability of selecting 2 consonants from the word HOME is given by:

1/6 = 0.1667

Therefore, the probability of not getting any vowel when 2 letters are taken at random from the word HOME is 0.1667.

Conclusion:
In conclusion, the probability of not getting any vowel when 2 letters are taken at random from the word HOME is 0.1667.

OK...total possibilities are 12... the letters with no vowels are 2(HM,MH)... so probability is 2/12= 1/6