A group of $n$ students doing an art project used red, blue, and yellow paint. Every student used at least one colour, and some students used more than one colour.

- The yellow paint was used by a total of 46 students.
- The red paint was used by a total of 69 students.
- The blue paint was used by a total of 104 students.
- Exactly 14 students used both yellow and blue and did not use red.
- Exactly 13 students used both yellow and red and did not use blue.
- Exactly 19 students used both blue and red and did not use yellow.
- Exactly 16 students used all three colours.

What is the value of $n$ ?
Solution 1

There are seven possibilities for the colours used by any individual student: they used exactly one colour (three possibilities), they used exactly two colours (three possibilities), and they used all three colours (one possibility).

We are given the number of students who used each combination of two colours and we are given the number of students who used all three colours.

To find the total, we need to determine how many students used only yellow, only red, and only blue. The partially-completed Venn Diagram below includes the given information:

![](https://cdn.mathpix.com/cropped/2024_04_17_698c2a84b151b65f3cc1g-09.jpg?height=635&width=594&top_left_y=623&top_left_x=820)

A total of 46 students used the yellow paint. Of these 46, 14 also used blue (and not red), 13 also used red (and not blue), and 16 also used both blue and red.

Therefore, the number of students who used only yellow paint is $46-14-13-16=3$.

By similar reasoning, the number of students who used only red paint is $69-13-19-16=21$ and the number of students who used only blue paint is $104-14-19-16=55$. The completed Venn diagram is below:

![](https://cdn.mathpix.com/cropped/2024_04_17_698c2a84b151b65f3cc1g-09.jpg?height=632&width=594&top_left_y=1668&top_left_x=820)

The total number of students is equal to the sum of the numbers in the seven regions of the Venn diagram, or

$$
3+21+55+14+13+19+16=141 .
$$

## Solution 2

Since every student used at least one colour, the total $46+69+104=219$ includes every
student, but it counts some students multiple times.

This total includes twice the number of students who used exactly two colours and three times the number of students who used all three colours.

If we subtract from 219 the number of students who used exactly two colours and twice the number of students who used all three colours, we will get the total number of students.

The number of students is $219-14-13-19-2(16)=141$.

ANSWER: 141