The letters of the word LOGARITHM are arranged at random. Find the probability that
a) Vowels are always together.
b) Vowels are never together.
c) Exactly 4 letters between G and H
d) Begin with O and end with T.
e) Start with vowel and end with consonant.
Answers
Answer:
1/12
11/12
1/108
1/72
1/4
Step-by-step explanation:
Hi,
Given that the letters of the word LOGARITHM are arranged at
random. Since there are 9 distinct letters, they can be arranged
in 9! ways.
a) Vowels are always together
There are 3 vowels, taking all 3 vowels together as 1 and the
rest 6 consonants, we can arrange in 7! ways , but 3 vowels
among themselves they can be arranged in 3! ways, total
number of ways are 7!*3!
Probability that vowels are always together is 7!*3!/9! = 1/12
b) Probability that Vowels are never together
It is same as 1 - Probability that vowels are always together
= 1 - 1/12 = 11/12
c) Exactly 4 letters between G and H:
For this we need to select 4 letters and treat G and those 4
letters and H as 1 and then this 6 letter group and the rest 3
letters can be arranged in 4! ways, G and H can interchange
their positions , this will be done in 2 ways and selection of 4
letters that could lie between G and H could be done in ⁸C₄
ways, hence total number of ways this arrangement could be
done are
4!*2*⁸C₄
So, the probability that exactly 4 letters between G and H will be
4!*2*⁸C₄ / 9! = 1/108
d) Begin with O and end with T:
So, there are 7 letters in between these O and T, which can be
permuted among themselves in 7! ways
Probability that word begin with O and end with T is 7!/9!
= 1/72
e) Start with vowel and end with consonant
Starting vowel can be chosen from 3 vowels in 3 distinct ways
and the end consonant could be chosen from 6 consonants in 6
distinct ways and the rest of 7 letters can be permuted among
themselves in 7! ways, so total number of ways this arrangement
could be done are 3*6*7!
So, the probability that words start with vowel and end with
consonant = 3*6*7!/9! = 1/4
Hope, it helps!