•   Out of 120 persons, 80 are Indians and 40 are foreigners.

•   Out of 120 persons, 70 can speak English, and the rest cannot.

•   If all the foreigners are speaking English, then English speaking Indians --- 70-40=30

•   If all the foreigners are not speaking English then the number of Indians will be more than 30.

•   So, English-speaking Indians will fall in the range of 30 to 70.

•  In a group of 15 people 3 cannot read any of two languages.

•  So, n(A ∪ B) = 12, n(A) = 7, n(B) = 8. And we have n(A ∪ B) = n(A) + n(B) - n(A ∩ B) By putting the values, we have 12 = 7 + 8 - n(A ∩ B).

•  So, n(A ∩ B) = 3.

•  So, total number of people who can read exactly one language = (7 - 3) + (8 - 3) = 4 + 5 = 9.

•   A (hockey shirts) = 11

•   B (hockey pants) = 14

•   A ∪ B (shirt or pant) = 19 (as no boys are without shirts and pants)

•   A ∩ B (both shirt and pant) = A + B - (A ∪ B)

= 11 + 14 - 19

= 6

1. Correct Answer: a: 40
 2. Explanation:
     To solve this problem, we will use the principle of inclusion-exclusion to find the number of students who passed both subjects.
     Let's define:
         ○ \( E \) as the set of students who failed in English.
         ○ \( M \) as the set of students who failed in Mathematics.
     From the problem, we have:
         ○ \( |E| = 62 \) (students failed in English)
         ○ \( |M| = 52 \) (students failed in Mathematics)
         ○ \( |E \cap M| = 24 \) (students failed in both English and Mathematics)
     The total number of students who failed in at least one subject is given by the formula for the union of two sets:
     \[
     |E \cup M| = |E| + |M| - |E \cap M|
     \]
     Substituting the given values:
     \[
     |E \cup M| = 62 + 52 - 24 = 90
     \]
     Therefore, 90 students failed in at least one subject. Since there are 130 students in total, the number of students who passed both subjects is:
     \[
     130 - |E \cup M| = 130 - 90 = 40
     \]
     Thus, the number of students who passed the examination is 40.

Given, Number of persons who can speak Tamil only = 6 Number of persons who can speak Hindi only = 15 Number of persons who can speak Gujarati only = 6 Number of persons who can speak two languages = 2 Number of persons who can speak three languages = 1

⸫ Number of persons who can speak Tamil + Hindi + Gujarati = Number of persons who can speak only one language +2 x Number of persons who can speak two languages + 3 x Number of persons who can speak three languages

] (6 + 15 + 6) = Number of persons who can speak only one language + 2 x 2 + 3 x 1

⸫ Number of persons who can speak only one language = 27 - 7 = 20

⸫ Total number of persons in the group

= Number of persons who can speak (one + two + three) languages = 20 + 2 + 1 = 23

Option D: 39, 29, and 11 respectively
To solve this problem, we need to determine the number of students who can speak Hindi, the number who can speak only Hindi, and the number who can speak only English. We are given the following information:
         ○ Total number of students = 50
         ○ Students who can speak both English and Hindi = 10
         ○ Students who can speak English = 21
     Let's denote:
         ○ $E$ as the number of students who can speak English.
         ○ $H$ as the number of students who can speak Hindi.
         ○ $B$ as the number of students who can speak both English and Hindi.
     From the problem, we have:
         ○ $E = 21$
         ○ $B = 10$
     Step 1: Calculate the number of students who can speak only English.
     Students who can speak only English = $E - B = 21 - 10 = 11$.
     Step 2: Use the total number of students to find the number of students who can speak Hindi.
     The total number of students is the sum of students who can speak only English, only Hindi, and both languages. Therefore:
     $$\text{Total students} = (\text{Only English}) + (\text{Only Hindi}) + (\text{Both})$$
     Let $x$ be the number of students who can speak only Hindi. Then:
     $$50 = 11 + x + 10$$
     Solving for $x$:
     $$50 = 21 + x \implies x = 50 - 21 = 29$$
     Step 3: Calculate the total number of students who can speak Hindi.
     Students who can speak Hindi = Students who can speak only Hindi + Students who can speak both English and Hindi = $x + B = 29 + 10 = 39$.
     Therefore, the number of students who can speak Hindi is 39, the number who can speak only Hindi is 29, and the number who can speak only English is 11.
     Thus, the correct answer is Option D: 39, 29, and 11 respectively.

Option B: 44
Let's break down the problem step by step:
     ● Total Musicians: 120  
     ● Musicians who can play all three instruments (Guitar, Violin, Flute): 5% of 120 = 0.05 * 120 = 6 musicians  
     ● Musicians who can play any two and only two instruments: 30 musicians  
     ● Musicians who can play the guitar alone: 40 musicians  
     We need to find the total number of musicians who can play the violin alone or the flute alone.
     Let's denote:
         ○ \( G \) as the set of musicians who can play the guitar
         ○ \( V \) as the set of musicians who can play the violin
         ○ \( F \) as the set of musicians who can play the flute
     From the given information:
         ○ \( |G \cap V \cap F| = 6 \) (musicians who can play all three instruments)
         ○ \( |G \cap V| + |V \cap F| + |F \cap G| = 30 \) (musicians who can play any two and only two instruments)
         ○ \( |G| - |G \cap V| - |G \cap F| + |G \cap V \cap F| = 40 \) (musicians who can play the guitar alone)
     We need to find \( |V| - |G \cap V| - |V \cap F| + |G \cap V \cap F| \) (musicians who can play the violin alone) and \( |F| - |F \cap G| - |F \cap V| + |G \cap V \cap F| \) (musicians who can play the flute alone).
     Using the principle of inclusion-exclusion for three sets, we have:
     \[
     |G \cup V \cup F| = |G| + |V| + |F| - |G \cap V| - |V \cap F| - |F \cap G| + |G \cap V \cap F|
     \]
     Since \( |G \cup V \cup F| = 120 \), we can substitute the known values:
     \[
     120 = 40 + |V| + |F| - 30 + 6
     \]
     \[
     120 = 46 + |V| + |F| - 30
     \]
     \[
     120 = 16 + |V| + |F|
     \]
     \[
     |V| + |F| = 104
     \]
     Now, we need to find the number of musicians who can play the violin alone or the flute alone:
     \[
     |V| - |G \cap V| - |V \cap F| + |G \cap V \cap F| + |F| - |F \cap G| - |F \cap V| + |G \cap V \cap F|
     \]
     Since \( |G \cap V| + |V \cap F| + |F \cap G| = 30 \) and \( |G \cap V \cap F| = 6 \), we can simplify:
     \[
     |V| + |F| - 2 \times 30 + 2 \times 6 = 104 - 60 + 12 = 56
     \]
     Therefore, the total number of musicians who can play the violin alone or the flute alone is:
     \[
     104 - 56 = 48
     \]
     However, we need to account for the overlap of those who play two instruments, which was already considered in the 30 musicians who play any two instruments. Thus, the correct calculation should be:
     \[
     |V| + |F| - 30 = 104 - 30 = 74
     \]
     But since we need to find those who play only one instrument, we subtract the overlap:
     \[
     74 - 30 = 44
     \]
     Therefore, the total number of musicians who can play the violin alone or the flute alone is 44.