To solve this problem, let's carefully break it into steps.

Total Club Members

The club consists of:

• 10 men
• 12 women
  Total members = $10 + 12 = 22$

Choosing Officers

The club needs to choose 4 officers: president, vice president, treasurer, and secretary. These positions are distinct, meaning the order in which they are chosen matters.

The number of ways to assign these 4 positions is: $P(22, 4) = \frac{22!}{(22-4)!} = \frac{22 \cdot 21 \cdot 20 \cdot 19}{1}$

$P(22, 4) = 175,560 \quad \text{(ways to choose the officers)}.$

Choosing Advisory Committee

After selecting the 4 officers, 18 members remain ($22 - 4 = 18$). The advisory committee is made up of 5 people, and since these positions are not distinct, the order doesn't matter. This is a combination problem.

The number of ways to choose 5 people out of 18 is: $C(18, 5) = \frac{18!}{5!(18-5)!} = \frac{18 \cdot 17 \cdot 16 \cdot 15 \cdot 14}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

$C(18, 5) = 8,568 \quad \text{(ways to choose the advisory committee)}.$

Total Number of Ways

Now, to find the total number of ways to select the officers and the committee, we multiply the results from the two steps: $\text{Total Ways} = P(22, 4) \cdot C(18, 5)$

$\text{Total Ways} = 175,560 \cdot 8,568 = 1,503,838,080.$

Final Answer:

The total number of ways to choose the officers and advisory committee is: $\boxed{1,503,838,080}$

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Would you like further clarification on any part of this calculation?

Relative Questions:

1. How does the calculation change if officers and committee members must include at least one man and one woman?
2. What happens if the committee selection allows officers to also serve as members?
3. How would this problem differ if all positions were chosen simultaneously rather than sequentially?
4. What is the result if the order of selecting committee members matters?
5. How can these results be generalized for clubs with different sizes?

Tip:

For problems involving both permutations and combinations, always calculate step-by-step, starting with the distinct positions before moving to groups where order doesn't matter.