The problem involves analyzing data from a survey of 140 wedding guests regarding their preferred entrees: chicken, beef, and fish. We are given information about how many people like each dish and combinations of dishes. We need to determine how many people like only fish and how many like none of the three dishes. We also need to fill out the Venn Diagram.

1. Problem Description

The problem involves analyzing data from a survey of 140 wedding guests regarding their preferred entrees: chicken, beef, and fish. We are given information about how many people like each dish and combinations of dishes. We need to determine how many people like only fish and how many like none of the three dishes. We also need to fill out the Venn Diagram.

2. Solution Steps

First, let's define some variables to represent the number of people in each region of the Venn diagram:

* $a$ = only chicken

* $b$ = only beef

* $c$ = only fish

* $d$ = chicken and beef only

* $e$ = chicken and fish only

* $f$ = beef and fish only

* $g$ = chicken, beef, and fish

* $h$ = none of the three

We are given the following information:

* Total number of people = 140

* Chicken = 90

* Beef = 70

* Fish = 90

* Chicken and Fish = 60

* Chicken and Beef = 40

* Beef and Fish = 40

* Chicken, Beef, and Fish = 20

From this, we can derive the following:

$g = 20$

$e + g = 60$, so $e = 60 - 20 = 40$

$d + g = 40$, so $d = 40 - 20 = 20$

$f + g = 40$, so $f = 40 - 20 = 20$

We also know:

$a + d + e + g = 90$ (Chicken)

$b + d + f + g = 70$ (Beef)

$c + e + f + g = 90$ (Fish)

And,

$a + b + c + d + e + f + g + h = 140$ (Total)

Now, we can solve for $a$, $b$, and $c$:

$a + 20 + 40 + 20 = 90$, so $a = 90 - 80 = 10$

$b + 20 + 20 + 20 = 70$, so $b = 70 - 60 = 10$

$c + 40 + 20 + 20 = 90$, so $c = 90 - 80 = 10$

To find $h$, we can plug the values we found into the total equation:

$10 + 10 + 10 + 20 + 40 + 20 + 20 + h = 140$

$130 + h = 140$

$h = 140 - 130 = 10$

a) How many like only fish?

$c = 10$

b) How many do not eat any of the three?

$h = 10$

c) Venn Diagram:

* Only Chicken: $a=10$

* Only Beef: $b=10$

* Only Fish: $c=10$

* Chicken and Beef only: $d=20$

* Chicken and Fish only: $e=40$

* Beef and Fish only: $f=20$

* Chicken, Beef and Fish: $g=20$

* None of the three: $h=10$

3. Final Answer

a) $10$

b) $10$