In a group of persons travelling in a bus, 6 persons can speak Tamil, 15 can speak Hindi and 6 can speak Gujarati. In that group none can speak any other language. If 2 persons in the group can speak two languages only and one person can speak all the three languages, then how many persons are there in the group?

The correct option is 23.

Let us solve this problem step-by-step using a Venn diagram approach to find the total number of persons in the group.

Let:
T be the set of persons who speak Tamil.
H be the set of persons who speak Hindi.
G be the set of persons who speak Gujarati.

According to the question, we have the total number of speakers for each language:
The number of persons who can speak Tamil, $n\left(T\right)=6$
The number of persons who can speak Hindi, $n\left(H\right)=15$
The number of persons who can speak Gujarati, $n\left(G\right)=6$

We are also given the intersections of these sets:
1. One person can speak all the three languages:
$n(T\cap H\cap G)=1$

2. Two persons can speak two languages only.
Let us represent the regions containing exactly two languages as:
- Tamil and Hindi only ($a$)
- Hindi and Gujarati only ($b$)
- Tamil and Gujarati only ($c$)
The sum of these regions is:
$a+b+c=2$

Let us denote the number of persons who speak only one language as:
- Tamil only ($x$)
- Hindi only ($y$)
- Gujarati only ($z$)

We can write equations for the total speakers of each language by summing up their constituent regions in the Venn diagram:
For Tamil: $x+a+c+1=6$ ⇒ $x+a+c=5$
For Hindi: $y+a+b+1=15$ ⇒ $y+a+b=14$
For Gujarati: $z+b+c+1=6$ ⇒ $z+b+c=5$

Now, let us add these three equations together:
$(x+a+c)+(y+a+b)+(z+b+c)=5+14+5$

Simplifying the left-hand side:
$(x+y+z)+2(a+b+c)=24$

Substitute the value of $a+b+c=2$ into the equation:
$(x+y+z)+2\left(2\right)=24$
$(x+y+z)+4=24$
$x+y+z=20$

So, the number of persons who speak only one language is 20.

Finally, the total number of persons in the group is the sum of those who speak exactly one language, exactly two languages, and all three languages:
Total persons = (Persons speaking only 1 language) + (Persons speaking only 2 languages) + (Persons speaking all 3 languages)
Total = $(x+y+z)+(a+b+c)+1$
Total = $20+2+1=23$

Therefore, the total number of persons in the group is 23.