Sure! Let's solve this step by step.
### Step 1: Define the Prices and Total Amount
1. The teenager has [tex]$100.
2. The prices of the tops are $[/tex]20 and [tex]$40.
### Step 2: Determine Possible Combinations
We'll find the combinations of tops that add up to exactly $[/tex]100.
#### Combinations:
- If she buys tops costing [tex]$20 each:
- $[/tex]100 / [tex]$20 = 5 tops can be bought.
- If she buys tops costing $[/tex]40 each:
- [tex]$100 / $[/tex]40 = 2.5 tops.
- Since she can't buy a half top, the maximum number of tops costing [tex]$40 she can buy is 2.
### Step 3: List Viable Combinations
Now let's list out all the possible combinations of tops costing $[/tex]20 and [tex]$40 that total exactly $[/tex]100.
1. [tex]\(1 \text{ top of } \$20 \text{ and } 2 \text{ tops of } \$40 \)[/tex]
2. [tex]\( 3 \text{ tops of } \$20 \text{ and } 1 \text{ top of } \$40 \)[/tex]
3. [tex]\( 5 \text{ tops of } \$20 \text{ and } 0 \text{ tops of } \$40 \)[/tex]
So the possible sequences are:
[tex]\[ (1, 2), (3, 1), (5, 0) \][/tex]
Where the first number represents the number of \[tex]$20 tops and the second number represents the number of \$[/tex]40 tops.
### Step 4: Determine the Specific Event
We need to find the event where she buys exactly 3 tops costing [tex]$20.
From the possible sequences, we can see:
\[ (3, 1) \]
This means that she can have exactly 3 tops costing $[/tex]20 and, in this scenario, she will also buy 1 top costing [tex]$40.
### Step 5: Draw the Tree (Conceptual)
To represent these possibilities in a tree form visually:
1. Start with $[/tex]100.
2. Create branches for each possible purchase:
- Branch 1: Buy a [tex]$20 top
- Branch 2: Buy a $[/tex]40 top
3. Continue branching out until the total expenditure reaches [tex]$100.
Example:
```
Initial: $[/tex]100
/ \
[tex]$80 $[/tex]60
/ \ / \
[tex]$60 $[/tex]40 [tex]$40 $[/tex]20
...continue until [tex]$0 is left
```
### Summary:
- The possible sequences of prices she could choose are \((1, 2), (3, 1), (5, 0)\).
- The event where she chooses exactly 3 tops costing $[/tex]20 is represented by the sequence [tex]\((3, 1)\)[/tex]. This indicates she buys 3 tops costing [tex]$20 and 1 top costing $[/tex]40.
