How do you do factorials on a TI-84 calculator, and what is the significance of factorial operations...

2. Enter the number for which you want to calculate the factorial. For example, if you want to calculate the factorial of 5, simply press `5`.
3. Press the `MATH` button, which is located below the `ALPHA` button and to the left of the `ARROW` buttons.
4. Use the arrow buttons to navigate to the `PRB` (probability) menu.
5. Select the `!` (factorial) function by pressing the number next to it or by using the arrow keys to highlight it and then pressing `ENTER`.
6. Press `ENTER` again to calculate the factorial of the number you entered.

For example, to calculate `5!`, you would enter:

```
5 [MATH] → [PRB] 4 [ENTER]
```

The calculator will display the result, which is `120` in this case.

Factorial operations are fundamental in combinatorics and probability calculations, particularly when dealing with permutations and combinations. Here's why:

- Permutations: A permutation is an arrangement of objects in a specific order. The number of permutations of `n` distinct objects taken `r` at a time is given by the formula:

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

This formula calculates how many different ways you can arrange `r` objects out of a total of `n`.

- Combinations: A combination is a selection of objects where the order does not matter. The number of combinations of `n` distinct objects taken `r` at a time is given by the formula:

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

This formula calculates how many different groups of `r` objects you can form from a total of `n`.

In both formulas, the factorial operation (`!`) is used to calculate the total number of ways to arrange a set number of objects. Factorials grow very rapidly with larger values of `n`, which is why they are so important in calculating probabilities for events with many possible outcomes. The TI-84 calculator's ability to compute factorials quickly and accurately makes it a valuable tool for students and professionals working in fields that involve combinatorics and probability.