​(A) How many ways can a 2​-person subcommittee be selected from a committee of 9 ​people?
​(B) How many ways can a president and vice dash president be chosen from a committee of 9 ​people?


Let's address each part of the question separately.

**Part (A): Selecting a 2-person subcommittee from a committee of 9 people**

This is a combinations problem because the order in which the two people are selected does not matter. The formula for combinations is given by:

\[ C(n, r) = \frac{n!}{r!(n-r)!} \]

Here, \( n = 9 \) and \( r = 2 \).

So, we calculate:

\[ C(9, 2) = \frac{9!}{2!(9-2)!} = \frac{9!}{2! \times 7!} \]

Breaking it down:

\[ 9! = 9 \times 8 \times 7! \]

Thus:

\[ C(9, 2) = \frac{9 \times 8 \times 7!}{2! \times 7!} = \frac{9 \times 8}{2 \times 1} = \frac{72}{2} = 36 \]

So, there are **36 ways** to select a 2-person subcommittee from a committee of 9 people.

**Part (B): Choosing a president and vice-president from a committee of 9 people**

This is a permutations problem because the order in which the president and vice-president are chosen matters. The formula for permutations is given by:

\[ P(n, r) = \frac{n!}{(n-r)!} \]

Here, \( n = 9 \) and \( r = 2 \).

So, we calculate:

\[ P(9, 2) = \frac{9!}{(9-2)!} = \frac{9!}{7!} \]

Breaking it down:

\[ 9! = 9 \times 8 \times 7! \]

Thus:

\[ P(9, 2) = \frac{9 \times 8 \times 7!}{7!} = 9 \times 8 = 72 \]

So, there are **72 ways** to choose a president and vice-president from a committee of 9 people.

Would you like more details or have any questions?

Here are 5 questions you might want to ask:
1. How do you differentiate when to use permutations versus combinations?
2. Can you explain the steps to simplify factorials in these types of problems?
3. How would the calculation change if we wanted to select a 3-person subcommittee?
4. What if the committee had 12 people instead of 9? How would the calculations change?
5. Can you give an example of a real-world situation where these types of calculations are useful?

**Tip:** For combination problems, use the formula \( C(n, r) \) where order does not matter, and for permutation problems, use \( P(n, r) \) where order does matter.

Explore how to calculate combinations and permutations when selecting subcommittees and choosing leadership roles from a committee of 9 people. Learn the formulas for combinations and permutations, along with step-by-step solutions. Suitable for high school students studying advanced mathematics.

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