Let me extract and analyze the text from the uploaded image. Then I will provide a detailed solution. PIL import Image import pytesseract

Load the image from the provided file path

image_path = "/mnt/data/1733855885649uvqi7sbv.jpg" img = Image.open(image_path)

Use Tesseract to extract text from the image

extracted_text = pytesseract.image_to_string(img) extracted_text### Extracted Problem Details:

The problem involves the selection of members from a board of 19 members, and it is divided into two parts:

1. Part (a): How many different slates of four members (a President, a Vice President, a Treasurer, and a Secretary) can be chosen?
   This involves assigning specific roles to four members, so the order in which members are chosen matters.

2. Part (b): How many delegations of four members (all members are equal) can be chosen?
   Here, the members are chosen as a group with no specific roles, so the order does not matter.

Let's solve these step by step.

Solution:

Part (a):

We are assigning specific roles to the four members, so this is a problem of permutations. The formula for permutations is:

$nP_r = \frac{n!}{(n - r)!}$

Where:

• $n = 19$ (total members)
• $r = 4$ (positions to fill)

The answer is expressed as $nP_r$: $19P_4 = \frac{19!}{(19 - 4)!}$

Part (b):

Here, we are choosing a group of four members without assigning roles, so this is a problem of combinations. The formula for combinations is:

$nC_r = \frac{n!}{r!(n - r)!}$

Where:

• $n = 19$ (total members)
• $r = 4$ (members to choose)

The answer is expressed as $nC_r$: $19C_4 = \frac{19!}{4!(19 - 4)!}$

Final Answers:

1. Part (a): $19P_4$
2. Part (b): $19C_4$

Would you like me to elaborate further on how to simplify these expressions or provide additional examples?

Follow-Up Questions:

1. What is the difference between permutations and combinations in practical scenarios?
2. Can you calculate $19P_4$ and $19C_4$ numerically?
3. How does the problem change if only three members were to be chosen?
4. What are the factorial values for $19!$ and $15!$ used in these calculations?
5. Can you explain why order matters in Part (a) but not in Part (b)?

Tip: Always double-check whether "order matters" in a selection problem—it determines whether to use permutations or combinations.