91Ó°ÊÓ

Combinations are a way to select items from a larger pool, where the order of selection does not matter. For instance, choosing 2 objects out of 4 objects \(a, b, c, d\) means we are interested in the unique groups of 2 items we can form. When considering combinations, remember the following points:

• The order of items does not matter. \(a, b\) is the same as \(b, a\).
• Combinations focus on the selection of items, not their arrangement.
• Each subset counts only once, regardless of the order of the elements.
• To list all combinations of a set, list all pairs, triples, etc., without repeating groups.

In our exercise, the combinations of \(a, b, c, d\) taken 2 at a time are: \(a, b\), \(a, c\), \(a, d\), \(b, c\), \(b, d\) and \(c, d\).

This method visually helps to see how unique sets are formed and avoids counting the same set more than once.

The binomial coefficient, often written as \(_{n}C_{r}\) or \(\binom{n}{r}\), represents the number of ways to choose \(r\) items from \(n\) items without regard to the order. It is a key concept in combinatorics and is calculated using the formula:

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

• \(n\) is the total number of items.
• \(r\) is the number of items to choose.
• \(!\) signifies factorial, which represents the product of all positive integers up to that number.

In our exercise, \(_{4}C_{2}\) is used to find the number of combinations of 2 items out of 4. Plugging in the values, we get:

\[_{4}C_{2} = \frac{4!}{2!(4-2)!} = \frac{4!}{2!2!} = \frac{4 \times 3 \times 2 \times 1}{2 \times 1 \times 2 \times 1} = 6 \]
The result shows there are 6 ways to choose 2 items out of 4, which matches our previous manual listing of combinations.

Factorial, denoted by an exclamation mark \(!\), is a mathematical operation used to find the product of all positive integers up to a certain number. For example:

• \(4! = 4 \times 3 \times 2 \times 1 = 24\)
• \(3! = 3 \times 2 \times 1 = 6\)
• \(2! = 2 \times 1 = 2\)

Factorials grow very quickly and are frequently used in combinatorial calculations, such as determining the number of ways to arrange or choose items. In our exercise, to find \(_{4}C_{2}\), we needed to calculate several factorials:

\[ 4! = 24 \]
\[ 2! = 2 \]
The combination formula \(\binom{n}{r} = \frac{n!}{r!(n-r)!} \) thus required us to compute these factorials to obtain the final result:

\[_{4}C_{2} = \frac{4!}{2!2!} = 6 \]
This confirms the approach and understanding of how factorials simplify the calculation of combinations and binomial coefficients.