Hello, azuresonata!

This is a tricky Venn diagram problem . . .

A group of students was asked to among the 3 bands: M, S, and H.

21 prefer M

23 prefer S

10 prefer M or H but not S

14 prefer M or S but not H

12 prefers S but not M or H but not M

3 prefers H but not M or S

8 prefers S but not M or H

17 prefers at least 2 of the bands

27 prefers 1 at most.

My teacher told me that there is another way to construct this diagram

which is way faster than using the cardinality of the given sets.

Does anyone know how? I think algebra is the way to go. Code:

* - - - - - - - - - - - *
| M |
| |
| s * - - - - - - - + - - - *
| | | |
| | | |
* - - - + - - - + - - - * | |
| | | | | |
| | t | u | v | w |
| * - - - + - - - | - - - * |
| | | |
| | x | S |
| y * - - - + - - - - - - - *
| |
| H |
* - - - - - - - - - - - *

We have the three "circles" for $\displaystyle M, S,\text{ and }H.$

Label the seven regions with $\displaystyle s,t,u,v,w,x,y.$

Then translate the given facts to equations.

$\displaystyle \begin{array}{ccc}n(M) = 21 & s + t + u + v \:=\:21 \\

n(S) = 23 & u + v + w + x \:=\:23 \\

n([M \cup H]\,\cap S') = 10 & s + t + y \:=\:10 \\

n([M \cup S] \cap H') = 14 & s + v + w \:=\:14 \\

n(S \cup H) \cap M') = 13 & w + x + y \:=\:13 \\

n(H \cap M' \cap S') = 3 & y \:=\:3 \\

n(S \cap M' \cap H') = 8 & w \:=\:8 \\

n(\text{at least 2}) = 17 & t + u + v + x \:=\:17 \\

n(\text{at most 1}) = 27 & s + w + y \:=\:27

\end{array}$

And solve the system.

(We already have two of the values.)

.