3. A school has 37 vacancies for teachers, out of which 22 are English Langugge, 20 for History and 17 for Fine Art, Of these vacancies 11 are for both English Language and Hiatory, 8 for licth History and Fine Art and 7 for Engish La nguage and Fine Art. Using a Venn diagram, find the number of teachers who must be able to teach: a.) all the three subject, b.) Fine Art only, c.) English Language and History but not Fine Art.

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UpStudy AI · Accepted Answer

Let $$ g $$ be the number of teachers who can teach all three subjects. Define the following:

- Let $$ a $$ be the number of teachers who can teach only English.
- Let $$ b $$ be the number of teachers who can teach only History.
- Let $$ c $$ be the number of teachers who can teach only Fine Art.
- Let $$ d $$ be the number of teachers who can teach exactly English and History.
- Let $$ e $$ be the number of teachers who can teach exactly History and Fine Art.
- Let $$ f $$ be the number of teachers who can teach exactly English and Fine Art.

We are given:
$$
\begin{array}{l}
\text{Total English vacancies: } a + d + f + g = 22 \
\text{Total History vacancies: } b + d + e + g = 20 \
\text{Total Fine Art vacancies: } c + e + f + g = 17 \
\text{Teachers for both English and History (including all three): } d + g = 11 \
\text{Teachers for both History and Fine Art (including all three): } e + g = 8 \
\text{Teachers for both English and Fine Art (including all three): } f + g = 7 \
\end{array}
$$

From the overlapping information, we have:
$$
\begin{aligned}
d &= 11 - g, \
e &= 8 - g, \
f &= 7 - g.
\end{aligned}
$$

Now express the counts for teachers who teach only one subject:
1. For English:
   $$
   a = 22 - \bigl[(d) + (f) + g\bigr] = 22 - \bigl[(11 - g) + (7 - g) + g\bigr].
   $$
   Simplify:
   $$
   a = 22 - \bigl[18 - g\bigr] = 4 + g.
   $$
2. For History:
   $$
   b = 20 - \bigl[(d) + (e) + g\bigr] = 20 - \bigl[(11 - g) + (8 - g) + g\bigr].
   $$
   Simplify:
   $$
   b = 20 - \bigl[19 - g\bigr] = 1 + g.
   $$
3. For Fine Art:
   $$
   c = 17 - \bigl[(e) + (f) + g\bigr] = 17 - \bigl[(8 - g) + (7 - g) + g\bigr].
   $$
   Simplify:
   $$
   c = 17 - \bigl[15 - g\bigr] = 2 + g.
   $$

The total number of vacancies equals the sum of all regions:
$$
a + b + c + d + e + f + g = 37.
$$

Substitute the expressions we derived:
$$
(4+g) + (1+g) + (2+g) + (11-g) + (8-g) + (7-g) + g = 37.
$$

Combine like terms:
$$
(4+1+2+11+8+7) + (g+g+g - g - g - g + g) = 37.
$$

Calculate the constant sum:
$$
4+1+2+11+8+7 = 33.
$$
For the $$ g $$ coefficients:
$$
g+g+g - g - g - g + g = g.
$$

Thus:
$$
33 + g = 37.
$$
Solve for $$ g $$:
$$
g = 37 - 33 = 4.
$$

Now answer the questions:

a.) The number of teachers who can teach all three subjects is 
$$
g = 4.
$$

b.) The number of teachers who can teach Fine Art only is 
$$
c = 2 + g = 2 + 4 = 6.
$$

c.) The number of teachers who can teach English and History but not Fine Art is 
$$
d = 11 - g = 11 - 4 = 7.
$$

Thus, the answers are:
- All three subjects: $$4$$.
- Fine Art only: $$6$$.
- English and History but not Fine Art: $$7$$.
- All three subjects: 4 teachers. - Fine Art only: 6 teachers. - English and History but not Fine Art: 7 teachers.

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