91Ó°ÊÓ

In probability and combinatorics, combinations are used to find how many ways you can choose items from a larger set without considering the order. Imagine you have a deck of cards, and you want to pick 2 cards. Combinations help us find out how many different pairs of cards we can pick.

The combination formula is given by: \[ { }_{n} C_{r} = \frac{n!}{r!(n-r)!} \] Where:

• \( n \) is the total number of items.
• \( r \) is the number of items to choose.

For example, in our exercise, we need to find: \[ { }_{10} C_{2} \]
This means we are choosing 2 items out of a total of 10.

To use the combination formula, we need to understand factorials. A factorial, represented by the symbol \(!\), is the product of all positive integers up to a specific number. For instance: \[ 10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3,628,800 \] This shows that the factorial of 10 (\(10!\)) is the result of multiplying 10 by all the smaller positive integers down to 1.

For the given exercise, we also need the factorial of 2 (\(2!\)): \[ 2! = 2 \times 1 = 2 \] And the factorial of \((10-2)\), which is 8 (\(8!\)): \[ 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40,320 \] These calculated factorials will then be used in our combination formula.

Combinatorial notation is a compact way to represent the number of combinations. The standard notation is \( { }_{n} C_{r} \), which stands for the number of ways to choose \( r \) items from \( n \) items without regard to order.

For our exercise, \[ { }_{10} C_{2} \] tells us to choose 2 items from a set of 10. We can simplify this as: \[ { }_{10} C_{2} = \frac{10!}{2!(10-2)!} \]
Let’s substitute our values: \[ { }_{10} C_{2} = \frac{10!}{2!⋅8!} \rightarrow \frac{3,628,800}{2 ⋅ 40,320} = 45 \] This means there are 45 different ways to choose 2 items out of 10.
Understanding combinatorial notation helps immensely when dealing with larger sets and more complex problems.