How to Use nCr, nPr and Factorial on a Calculator

Three buttons on a scientific calculator confuse more students than almost any other group: the factorial key, nPr, and nCr. Each one answers a different counting question, and mixing them up gives a wrong answer even when every digit you typed was correct. Once you see the exact formula behind each button, the three stop looking alike.

What Does Factorial Mean on a Calculator?

Factorial is a math operation that multiplies a whole number by every positive whole number smaller than it, down to 1. The calculator symbol is an exclamation point, so 5! means “5 factorial.”

• n = the whole number you start from (must be 0 or a positive integer)
• n! = “n factorial,” the product of every whole number from n down to 1

Factorial grows fast because each new term multiplies the running total instead of adding to it. Three examples show the pattern clearly.

1. \( 3! = 3 \times 2 \times 1 = 6 \)
2. \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)
3. \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)

A special case worth memorizing is \( 0! = 1 \), not zero. Mathematicians define it that way so the nPr and nCr formulas below still work correctly when r equals n.

On most scientific calculators, factorial sits behind a single key, often labeled x! or n!. Type the number first, then press the factorial key, and the full product appears instantly without multiplying each step by hand.

How Do You Calculate nPr, a Permutation, on a Calculator?

A permutation counts the number of ways to arrange r items chosen from a group of n items, where the order of the result matters. The calculator function is nPr, sometimes shown as P(n,r).

• n = the total number of items available to choose from
• r = the number of items you select and place in order
• nPr = the number of distinct ordered arrangements possible

Take five people racing for two medals, first place and second place. That question asks for 5P2, since the two winning spots have a distinct order.

1. Write the formula: \( nPr = \frac{n!}{(n-r)!} \)
2. Plug in n = 5 and r = 2: \( 5P2 = \frac{5!}{(5-2)!} \)
3. Calculate each factorial: \( 5! = 120 \) and \( (5-2)! = 3! = 6 \), so \( 5P2 = \frac{120}{6} \)
4. Divide to get the answer: \( 5P2 = 20 \)

Twenty different first-and-second-place orderings are possible among five racers, and the calculator confirms it in one keypress once you enter the right values.

A larger example shows the same pattern. Eight runners competing for three medals gives \( 8P3 = \frac{8!}{5!} = 336 \), since swapping which runner takes gold, silver, or bronze always creates a new, distinct outcome.

How Do You Calculate nCr, a Combination, on a Calculator?

A combination counts the number of ways to choose r items from a group of n items when order does not matter. The calculator function is nCr, sometimes shown as C(n,r) or written as a binomial coefficient.

• n = the total number of items available to choose from
• r = the number of items selected, with no regard to order
• nCr = the number of distinct groups possible

Take the same five people, but this time picking a 2-person cleanup crew instead of ranked medals. Swapping the two names in the group does not create a new crew, so this question asks for 5C2.

1. Write the formula: \( nCr = \frac{n!}{r!(n-r)!} \)
2. Plug in n = 5 and r = 2: \( 5C2 = \frac{5!}{2!\,(5-2)!} \)
3. Calculate each factorial: \( 5! = 120 \), \( 2! = 2 \), and \( 3! = 6 \), so the denominator is \( 2 \times 6 = 12 \)
4. Divide to get the answer: \( 5C2 = \frac{120}{12} = 10 \)

Ten distinct 2-person crews can form from five people, half as many outcomes as the 20 ranked pairs from the permutation example, because nCr always removes the extra orderings that nPr counts separately.

The larger example follows the same rule. Choosing a 3-person committee from eight candidates gives \( 8C3 = \frac{8!}{3!\,5!} = 56 \), far fewer than the 336 ranked outcomes for the same eight people and three spots.

Permutation or Combination: Which One Should You Use?

Use a permutation whenever the positions or roles in the outcome are distinct from each other. Use a combination whenever the outcome is just a set, with no roles attached to the members.

Picture a stack of index cards, each with a name on it. Dealing three cards into three labeled slots, first, second, and third, is a permutation problem, since each slot is a distinct role. Dealing three cards into one plain pile is a combination problem, since the pile has no internal order at all.

Ranking runners for 1st, 2nd, and 3rd place is a permutation, because swapping two runners between two medal spots creates a genuinely different result. The same eight runners choosing a 3-person relay team, with no assigned running order, is a combination instead.

A simple test settles most cases quickly. Ask whether swapping two selected items changes the outcome. A “yes” answer points to nPr; a “no” answer points to nCr.

Lottery number draws, seating charts with named seats, and podium finishes are classic permutation situations. Committee selections, card hands, and pizza toppings picked as a set are classic combination situations.

How Do You Enter Factorial, nPr and nCr on a Real Calculator?

The Advanced Scientific Calculator on the MultiCalculators homepage includes dedicated x!, nCr, and nPr keys, so you never have to expand the factorials by hand.

1. Type the value of n first, such as 5.
2. Press the nPr or nCr key on the calculator.
3. Type the value of r, such as 2.
4. Press equals to see the final result, 20 for 5P2 or 10 for 5C2.

For a plain factorial, type the number, then press the x! key directly, with no second value needed. Parentheses rarely matter for a single factorial, but they help when a factorial sits inside a larger expression, a pattern covered in the sibling article on using parentheses on a scientific calculator.

What Mistakes Do People Make With Factorial, nPr and nCr?

The most common mistake is using nPr when the question actually describes a combination, which inflates the answer by a factor of r!. A committee question solved as a permutation gives a number far larger than the real count of distinct groups.

A second mistake is forgetting that 0! equals 1, not 0. Plugging a zero factorial into either formula as if it were zero breaks the division and produces a wrong or undefined result.

A third mistake is entering n and r in the wrong order on the calculator, since nPr and nCr always expect the total group size first and the chosen amount second.

• Using nPr for a question where order genuinely does not matter.
• Treating 0! as zero instead of 1.
• Swapping the n and r values when typing them into the calculator.
• Trying to take the factorial of a negative number or a non-whole number without a scientific mode built for it.

A fourth mistake worth flagging is assuming a large factorial always fits on the screen. Calculators display very large factorials in scientific notation once the digit count grows past the screen width, which can look like an error to someone expecting a plain integer.

Checking any unfamiliar result against a manual step-by-step calculation, using the method in the sibling article on verifying a calculator’s answer, catches most of these mistakes before they cause real problems on a test or a report.

FAQs About nCr, nPr and Factorial on a Calculator

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This article is for general math education only. Exact calculator behavior varies by brand and model, so check your device’s manual for specifics. Reviewed for accuracy by Prof. Dr. Khalil Mudassar, PhD. Last updated September 23, 2026.