1. Representation in Venn diagram

This is a problem given to us in our 3rd Year High school class.

A group of students was asked to choose the 3 bands namely: MCR, SC and H. 21 prefers MCR, 23 prefers SC, 10 prefers MCR or H but not SC, 14 prefers MCR or SC but not H, 12 prefers SC but not MCR or H but not MCR, 3 prefers H but not MCR or SC, 8 likes SC but not MCR or H. 17 likes at least 2 of the bands and 27 prefers 1 at most.

Can someone represent this in Venn Diagram?

Also, 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?

2. 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 $M, S,\text{ and }H.$
Label the seven regions with $s,t,u,v,w,x,y.$
Then translate the given facts to equations.

$\begin{array}{ccc}n(M) = 21 & s + t + u + v \:=\:21 \\
n(S) = 23 & u + v + w + x \:=\:23 \\
n([M \cup H]\,\cap S') = 10 & s + t + y \:=\:10 \\
n([M \cup S] \cap H') = 14 & s + v + w \:=\:14 \\
n(S \cup H) \cap M') = 13 & w + x + y \:=\:13 \\
n(H \cap M' \cap S') = 3 & y \:=\:3 \\
n(S \cap M' \cap H') = 8 & w \:=\:8 \\
n(\text{at least 2}) = 17 & t + u + v + x \:=\:17 \\
n(\text{at most 1}) = 27 & s + w + y \:=\:27
\end{array}$

And solve the system.

(We already have two of the values.)
.