A group consists of a number of students and each student of the group can speak at least one of the languages Bengali, Hindi and English. 65 can speak Bengali, 54 Hindi and 37 English; 31 can speak both Bengali and Hindi;17 both Hindi and English, and 18 both Bengali and English. Determine the greatest and least number of students in the group.?

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Problem:

A group consists of a number of students and each student of the group can speak at least one of the languages Bengali, Hindi and English. 65 can speak Bengali, 54 Hindi and 37 English; 31 can speak both Bengali and Hindi;17 both Hindi and English, and 18 both Bengali and English. Determine the greatest and least number of students in the group? Explain in details.

Solution:

Let's use Venn diagrams to solve the problem. Let B, H, and E represent the sets of students who speak Bengali, Hindi, and English, respectively. The numbers given in the problem can be represented in the following Venn diagram:

![image.png](attachment:image.png)

Finding the number of students who can speak all three languages:

To find the number of students who can speak all three languages, we can use the formula:

n(B ∩ H ∩ E) = n(B) + n(H) + n(E) - 2n(B ∩ H) - 2n(H ∩ E) - 2n(B ∩ E) + 3n(B ∩ H ∩ E)

Substituting the values from the problem, we get:

n(B ∩ H ∩ E) = 65 + 54 + 37 - 2(31) - 2(17) - 2(18) + 3n(B ∩ H ∩ E)

n(B ∩ H ∩ E) = 19 + 3n(B ∩ H ∩ E)

n(B ∩ H ∩ E) = 19 + 3n

where n is the number of students who can speak all three languages.

Finding the greatest number of students:

The greatest number of students in the group can be found by adding up the number of students who speak each language, subtracting the number of students who speak two languages, and adding the number of students who speak all three languages:

Greatest number of students = n(B) + n(H) + n(E) - n(B ∩ H) - n(H ∩ E) - n(B ∩ E) + n(B ∩ H ∩ E)

Substituting the values from the problem, we get:

Greatest number of students = 65 + 54 + 37 - 31 - 17 - 18 + n(B ∩ H ∩ E)

Greatest number of students = 90 + n(B ∩ H ∩ E)

The maximum value of n(B ∩ H ∩ E) is 19 (when all 19 students who can speak all three languages are included), so the greatest number of students in the group is:

Greatest number of students = 90 + 19 = 109

Finding the least number of students:

The least number of students in the group can be found by adding up the

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