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learnmgcat

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 #4
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+1

Here is part (b):

 

The number of ways to choose three subsets A, B, C of S is 2^n * 2^n * 2^n = 2^(3n).

 

To find the number of ways that A is a subset of B, and B is a subset of C, we can use the following approach:

 

Choose a subset C of S. There are 2^n ways to do this.

 

Choose a subset B of C. There are 2^|C| ways to do this, where |C| is the size of C.

 

Choose a subset A of B. There are 2^|B| ways to do this, where |B| is the size of B.

 

Therefore, the number of ways that A is a subset of B, and B is a subset of C is:

 

∑_{C⊆S} 2^|C| * 2^|B|

 

where the sum is over all subsets C of S.

 

We can simplify this expression by noting that for any fixed subset C of S, the number of ways to choose subsets B and A of C such that A is a subset of B is 2^(|C|+1). This is because we can choose any subset B of C, and then choose any subset A of B. There are 2^(|C|+1) ways to do this.

 

Therefore, the number of ways that A is a subset of B, and B is a subset of C is:

 

∑_{C⊆S} 2^(|C|+1)

 

We can evaluate this sum using the following trick:

 

∑_{C⊆S} 2^(|C|+1) = 2 * ∑_{C⊆S} 2^|C|

 

The sum on the right-hand side is just the number of subsets of S, which is 2^n. Therefore, we have:

 

∑_{C⊆S} 2^(|C|+1) = 2 * 2^n = 2^(n+1)

 

Finally, the probability that A is a subset of B, and B is a subset of C is:

 

2^(n+1) / 2^(3n) = 1 / 2^(2n-1)

 

Therefore, the probability that A is a subset of B, and B is a subset of C is 1 / 2^(2n-1).

Sep 24, 2024
 #1
avatar+1310 
0

To solve this problem, we need to understand the relationships within the triangle STU. Here are the steps to find the length of \( SX \):

 

### Step 1: Analyzing the Triangle


Given:


- \( S \), \( T \), and \( U \) are the vertices of the triangle.


- \( M \) is the midpoint of \( ST \).


- \( N \) is a point on \( TU \) such that \( SN \) is the altitude of the triangle.


- \( ST = SU = 13 \), \( TU = 8 \).


- \( UM \) and \( SN \) intersect at \( X \).

 

### Step 2: Applying the Median and Altitude Properties


Since \( M \) is the midpoint of \( ST \), \( SM = MT = \frac{13}{2} = 6.5 \).

 

Also, \( SN \) is an altitude, so it is perpendicular to \( TU \).

 

### Step 3: Use the Property of the Centroid


In any triangle, the centroid (intersection of the medians) divides each median in a 2:1 ratio. Since \( X \) is the intersection of the medians \( SN \) and \( UM \), it is the centroid of triangle \( STU \).

 

This implies:


\[
SX = \frac{2}{3} \times SN
\]


where \( SN \) is the altitude from \( S \) to \( TU \).

 

### Step 4: Calculate SN Using the Area of the Triangle


We use the fact that the area of the triangle can be calculated in two ways:


1. Using base \( TU \) and height \( SN \).


2. Using Heron's formula.

 

#### Heron's Formula:


First, calculate the semi-perimeter \( s \):


\[
s = \frac{ST + SU + TU}{2} = \frac{13 + 13 + 8}{2} = 17
\]


Then, calculate the area \( \Delta \):


\[
\Delta = \sqrt{s(s - ST)(s - SU)(s - TU)} = \sqrt{17(17 - 13)(17 - 13)(17 - 8)} = \sqrt{17 \times 4 \times 4 \times 9} = \sqrt{2448} = 24
\]

 

#### Area Using Altitude \( SN \):


The area can also be written as:


\[
\Delta = \frac{1}{2} \times TU \times SN = \frac{1}{2} \times 8 \times SN = 4 \times SN
\]


Equating the two expressions for the area:


\[
24 = 4 \times SN \implies SN = 6
\]

 

### Step 5: Calculate SX


Now that we know \( SN = 6 \), the length of \( SX \) is:


\[
SX = \frac{2}{3} \times 6 = 4
\]

 

Thus, the length of \( SX \) is \( \boxed{4} \).

Aug 14, 2024