The image contains two exercises related to combinatorics. Here’s a breakdown of each:

Exercise 9.25:

A ship carries 48 flags in total, with 12 flags of each color: red, white, blue, and black. Twelve of these flags are placed on a vertical pole to signal other ships. The question has two parts:

(a) Question:

How many of these signals use an even number of blue flags and an odd number of black flags?

To solve:

1. Let $x, y, z, w$ represent the number of red, white, blue, and black flags used, respectively, with $x + y + z + w = 12$.
2. Conditions:
   • $z$ (number of blue flags) must be even.
   • $w$ (number of black flags) must be odd.

(b) Question:

How many of these signals have at least three white flags or no white flags at all?

To solve:

1. Count the cases where $y \geq 3$ or $y = 0$, with $x + y + z + w = 12$.
2. Use combinations based on these constraints.

Exercise 9.26:

Assign 11 new employees to 4 subdivisions, with each subdivision receiving at least one new employee. Additionally, each subdivision can receive employees anywhere between 1 to 8 times.

To solve:

1. This problem can be addressed using combinatorial methods or generating functions.
2. The constraint of at least one employee per subdivision affects the distribution choices.

Would you like a detailed solution to any specific part?

Additional Practice Questions

1. How many ways can 12 flags be arranged if each color must appear exactly 3 times?
2. In how many ways can we select 12 flags out of 48 if no two flags of the same color can be adjacent?
3. How many ways are there to assign 11 employees to 4 subdivisions if no more than 5 employees can be in a single subdivision?
4. What is the probability that a randomly selected signal will contain at least one flag of each color?
5. How many arrangements of flags would contain exactly 6 blue flags?

Tip

When faced with combinatorial problems, identify if each choice is dependent or independent—this helps simplify calculations using combinatorics or generating functions.