Okay, here's the full treatment.

(Archie Meade beat me to it . . .)

(1) Rye has 16 suitors

(2) 8 are tall

(3) 9 are dark

(4) 5 are not tall nor handsome

(5) 8 have at least 2 qualities

(6) 10 are not handsome

(7) 1 is tall but not dark nor handsome.

(8) 3 are dark but not tall nor handsome.

How many are neither tall, dark nor handsome?

If Rye goes for physical appearance, will it be hard to choose among her suitors?

We have this three-ring Venn diagram.

Code:

*---------------------------*
| |
| *---------------* |
| | Tall | |
| | a *-------+---* |
| | | b | | |
| | *---+---* | c | |
| | | d | e | | | |
| *---+---+---+---* | |
| | | | | |
| | f | g | Dark | |
| | *---+-------* |
| | Hsome | |
| *-------* h |
| |
*---------------------------*

From statement (7): $\displaystyle a = 1$

From statement (8): $\displaystyle c = 3$

From statement (4): $\displaystyle c+h \,=\,5 \;\;\Rightarrow\;\; h = 2$

From statement (6): $\displaystyle a + b + c + h \,=\,10 \;\;\Rightarrow\;\; 1+b+3+2\,=\10 \;\;\Rightarrow\;\; b = 4$

The diagram becomes:

Code:

*---------------------------*
| |
| *---------------* |
| | Tall | |
| | 1 *-------+---* |
| | | 4 | | |
| | *---+---* | 3 | |
| | | d | e | | | |
| *---+---+---+---* | |
| | | | | |
| | f | g | Dark | |
| | *---+-------* |
| | Hsome | |
| *-------* 2 |
| |
*---------------------------*

$\displaystyle \begin{array}{ccccccccc}\text{From statement (2):} & d + e + 1 + 4 \:=\:8 & \Rightarrow & d + e &=& 3 & (a) \\ \text{From statement (3):} & e + f + 4 + 3 \:=\:9 & \Rightarrow & e + f &=& 2 & (b) \\ \text{From statement (5):} & d + e + f + 4 \:=\:8 & \Rightarrow & d + e + f &=& 4 & (c) \end{array} $

$\displaystyle \begin{array}{ccccccc}\text{Substitute (a) into (c):} & 3 + f \:=\:4 & \Rightarrow & f \:=\:1 \\ \text{Substitute (b) into (c):} & d + 2 \:=\:4 & \Rightarrow & d \:=\:2 \end{array}$

And hence: .$\displaystyle e \:=\:1\,\text{ and }\,g \:=\:2$

And the final diagram is:

Code:

*---------------------------*
| |
| *---------------* |
| | Tall | |
| | 1 *-------+---* |
| | | 4 | | |
| | *---+---* | 3 | |
| | | 2 | 1 | | | |
| *---+---+---+---* | |
| | | | | |
| | 2 | 1 | Dark | |
| | *---+-------* |
| | Hsome | |
| *-------* 2 |
| |
*---------------------------*

Define "hard to choose".

Among the 16 suitors, only 2 are short, light and ugly.

She has 14 chances out of 16 to seclecting a suitor

. . who is has at least one attractive characteristic.

If she insists on Perfection, only one suitor is tall **and** dark **and** handsome.