91Ó°ÊÓ

In statistics, the combination formula helps us determine the number of ways to choose a subset of items from a larger set. It's essential when the order of selection doesn’t matter. This is different from permutations, where order does count. The combination formula is represented as: \ \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \), where \(n\) is the total number of items, and \(r\) is the number of items to choose. In the given problem, \(n = 21\) and \(r = 9\). This calculates to: \ \[ \binom{21}{9} = \frac{21!}{9!(21-9)!} = \frac{21!}{9!12!} \].
This formula might look complex at first, but understanding each part makes it easier. The exclamation mark (!) denotes a factorial, which we'll explain next.

Factorials are a fundamental part of the combination formula. A factorial, denoted by an exclamation mark (!), represents the product of all positive integers up to a given number.
For example:

• \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
• \( 3! = 3 \times 2 \times 1 = 6 \)
• \( 1! = 1 \)
\( 0! = 1 \) by definition.
In our problem, we'd need to compute \( 21! \), \( 9! \), and \( 12! \). Computing these directly can be cumbersome due to their large size, so using a calculator or a math tool is helpful.
Plugging these into the combination formula: \ \[ \binom{21}{9} = \frac{21!}{9!12!} \], simplifies to 293,930 ways to choose 9 members out of 21 for the subcommittee.

Subsets are a smaller group of elements selected from a larger group. In combinatorics, subsets are comprised of elements chosen without regard to order. This makes combinations an ideal way to calculate the number of possible subsets.
For the Subcommittee on Economic Policy example, we are choosing a subset of 9 members out of a total of 21. Each potential group is a subset of the original committee.
It's important to note that if order did matter, we would be dealing with permutations instead of combinations. But here, since we only care about which members are chosen, not the sequence in which they are selected, combinations apply.
Using the combination formula, we find that there are exactly 293,930 possible subsets (committee structures) for the subcommittee. This ensures we account for all different groups that can be formed without considering order.