TI-84 Factorials (Graphing Calculator Guide)

To calculate a factorial on a TI‑84, enter a nonnegative whole number, choose the factorial command from the MATH menu’s PRB section, then evaluate. For example, (5! = 120). Use nPr for ordered selections and nCr for unordered selections. Check your setup and the calculator’s range: it can represent (69!), but not (70!).

Factorials can save time in counting and probability problems, but the calculator cannot tell whether you chose the right counting rule. A little setup is worth the effort: write down what is being counted, choose a factorial, permutation, or combination, and then enter the expression. This guide walks through the keys, limits, examples, and checks that help prevent avoidable mistakes.

TI‑84 Factorial Function: Menu Location and Calculator Compatibility

The factorial command is in the calculator’s probability menu, usually reached through MATH, then PRB. Menu positions can vary by model or operating-system version, so look for the displayed ! label rather than relying on a menu number. The method below applies to TI‑84 models with the standard menu layout.

First, test the command with a small value. On the home screen, enter 5, press MATH, move to PRB, select !, and press ENTER to place the symbol after the 5. Press ENTER again to calculate. The result should be 120.

The two ENTER presses have different jobs. The first selects the menu command and places it in the expression; the second evaluates that expression. If your model’s menu looks different, use the arrow keys to find PRB and read the command labels on screen.

A quick test helps confirm that you selected the factorial command, not nPr or nCr. If the expression on the home screen does not look like 5! before you evaluate it, clear it and try again.

Key takeaway: Check the screen before calculating. You should see the number followed by ! for a factorial.

Factorial Rules, Input Limits, and Related Counting Functions

A factorial is the product of a nonnegative whole number and every positive whole number below it: (n! = n(n-1)\cdots 1). By definition, (0! = 1). On a TI‑84, factorials are for nonnegative integers, and the calculator’s range limits how large a result it can show.

For example, (4! = 4 \times 3 \times 2 \times 1 = 24). The exclamation mark is a math symbol, not a punctuation mark or an exponent. So 4! means 24, not (4) raised to some power.

Use factorials only with whole-number inputs that are zero or greater. Do not enter a negative number or a decimal and expect the factorial key to calculate a general result. If a problem gives a non-integer or negative input, check the problem’s assumptions and the method your course expects.

The calculator’s range

The TI‑84 can represent (69!), which is about (1.7112 \times 10^{98}). It cannot represent (70!), which is about (1.1979 \times 10^{100}), because that exceeds its numeric range. Large values may appear in scientific notation, where E98 means “times (10^{98}).”

This limit matters even when a problem’s inputs are valid whole numbers. A factorial can be mathematically defined but too large for the calculator to display. If an entry gives an error or an unusable result, check the input and the size of the answer.

Choose between factorial, nPr, and nCr

A permutation, written (nPr), counts ways to select and arrange (r) items from (n) distinct items. A combination, written (nCr), counts ways to select (r) items when order does not matter. Both require whole numbers with (0 \le r \le n).

On the calculator, select nPr or nCr from the PRB menu. Enter the first value, the function, and then the second value. For example, 8 nCr 3 should produce 56. If the menu order differs on your calculator, select the function by its displayed label.

Key takeaway: Decide whether order matters before choosing a command. A correct calculation with the wrong function still gives the wrong answer to the problem.

Worked Examples and Error-Checking Strategies

Worked examples show how to turn a question into a calculator expression. The calculator handles the arithmetic; your written setup shows why that expression fits. I recommend keeping both visible in your work, especially when a problem includes several steps or asks for an explanation.

Example 1: Arrange every item

How many ways can six distinct items be arranged in a row? All six are used, and changing their order creates a new arrangement. The setup is (6!), so enter 6, select ! from MATH → PRB, and evaluate. The answer is 720.

Example 2: Choose and rank a group

A club has eight members and needs a president, vice president, and secretary. The roles differ, so who gets which role matters. Use (8P3), not (8C3). Enter 8 nPr 3; the result is 336.

A useful check is to compare the size of the answer with the unordered selection. There are only (8C3 = 56) groups of three, but each group can fill the three distinct roles in (3! = 6) ways. Since (56 \times 6 = 336), the permutation result is consistent.

Example 3: Choose a group where order does not matter

Suppose a teacher selects three students from a group of eight for a committee with no assigned roles. The order of selection does not change the committee, so use (8C3). Enter 8 nCr 3 and evaluate: the result is 56.

Example 4: Check the zero case

A problem may include (0!), particularly in a formula or a counting argument. The correct value is 1, not 0. Enter 0, choose !, and evaluate to confirm the result. This special definition keeps common formulas consistent.

A short error-checking routine

Before accepting an answer, work through these checks:

• Check the model: Does the problem ask for all arrangements, ordered selections, or unordered selections?
• Check the inputs: Are (n) and (r) nonnegative whole numbers? For nPr and nCr, is (r \le n)?
• Check the expression: Does the display show the intended values and command?
• Check the size: Is the factorial within the calculator’s range?
• Check the result: Can you confirm it with a formula, a smaller case, or another valid method?

A calculator result does not prove that the setup is right. For example, (8P3) and (8C3) both produce valid answers, but they answer different questions. Write the formula and what each variable means alongside your calculator work.

Key takeaway: Treat the calculator as an arithmetic tool, not as a substitute for identifying the counting model.

Applications in Probability, Combinatorics, and Further Study

Factorials appear in counting problems and in formulas used for probability. They help count arrangements and selections, but the meaning of the question still determines which expression to use. Build the setup first, then use the TI‑84 to evaluate it and check that the answer fits the situation.

In combinatorics, (n!) counts the arrangements of (n) distinct objects. The nPr function applies when you select some objects and assign an order or distinct positions. The nCr function applies when only the chosen group matters.

In probability, combinations often count equally likely selections. For instance, if you are counting possible groups drawn from a larger set, nCr may give the number of groups. To find a probability, you also need a count of the favorable outcomes and the total outcomes, using a model that matches the problem. The calculator’s nCr result alone is not the probability.

A practical workflow is:

• Translate the wording into “arrange all,” “choose and order,” or “choose without order.”
• Write the matching expression, such as (n!), (nPr), or (nCr).
• Enter it using the PRB menu and check the screen.
• Compare the result with a formula or a simple count where possible.
• Include the expression and variable meanings in your written solution.

For further study, the same formulas may appear in algebra, probability, and statistics. Getting comfortable with the difference between permutations and combinations makes those problems easier to read. It also helps you catch answers that are numerically valid but do not match the question.

Key takeaway: Use factorial commands to evaluate a setup you understand. Keep the reasoning with the result.

Frequently Asked Questions

These quick answers cover common questions about the factorial key, valid inputs, related functions, and calculator limits. If your TI‑84 menu looks different, follow the labels shown on your screen rather than assuming every model has the same menu positions.

Where is the factorial button on a TI‑84?
It is in the MATH menu under PRB. Select the displayed ! command.

How do I calculate (5!) on a TI‑84?
Enter 5, choose ! from MATH → PRB, then press ENTER to calculate. The result is 120.

What does (0!) equal?
(0! = 1). This is the standard definition of zero factorial.

Can I calculate a factorial for a decimal?
No. The TI‑84 factorial command is for nonnegative whole-number inputs, not arbitrary decimals.

Can I calculate the factorial of a negative number?
No. Factorials on the TI‑84 require nonnegative integer inputs.

What is the largest factorial the TI‑84 can represent?
The calculator can represent (69!). The value of (70!) exceeds its numeric range.

What is the difference between nPr and nCr?
Use nPr when order matters, such as assigning people to different roles. Use nCr when order does not matter, such as selecting a committee.

What input rules apply to nPr and nCr?
Use nonnegative whole numbers, with (r \le n). The first input is (n), and the second is (r).

Why does my menu not match the instructions?
TI‑84 model and operating-system menus can vary. Open MATH, find PRB, and select the function by its displayed label.

Why is my calculator answer not enough for my homework?
The result only evaluates the expression you entered. Show the formula and explain why factorial, nPr, or nCr fits the problem.

(This article was written by one of our staff writers, Christopher Langston. Visit our Meet the Team page.)