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

Let the sets of people speaking French, Spanish, and English be denoted as $F$, $S$, and $E$ respectively.
Given:

* Total speaking French, $n(F)=6$
* Total speaking Spanish, $n(S)=15$
* Total speaking English, $n(E)=6$
* Number of people speaking exactly three languages, $n(F\cap S\cap E)=1$
* Number of people speaking exactly two languages $=2$

Using the principle of inclusion-exclusion for three sets to find the total unique individuals $n(F\cup S\cup E)$:

$$\text{Total individuals}=\sum \text{Only 1 language}+\sum \text{Exactly 2 languages}+\sum \text{Exactly 3 languages}$$

Let us count total language instances: $n(F)+n(S)+n(E)=6+15+6=27$.
In this sum ($27$):

* People speaking exactly 1 language are counted once.
* People speaking exactly 2 languages are counted twice.
* People speaking exactly 3 languages are counted thrice.

Therefore:

$$\text{Total language instances}=(\text{Exactly 1})+2\times (\text{Exactly 2})+3\times (\text{Exactly 3})$$

$$27=(\text{Exactly 1})+2(2)+3(1)$$

$$27=(\text{Exactly 1})+4+3$$

$$\text{Exactly 1 language}=27-7=20$$

Now, total number of persons in the group is:

$$\text{Total persons}=(\text{Exactly 1})+(\text{Exactly 2})+(\text{Exactly 3})=20+2+1=23$$