Combination on Scientific Calculator: How to Do nCr

To calculate a combination (nCr) on a scientific calculator, use the nCr function, usually accessed by pressing SHIFT or 2ndF followed by the division key (÷). Enter the total number of items (n), press the nCr key, enter the number chosen (r), then press equals. For example, to find how many 3-item groups can be made from 10 items, press 10 SHIFT ÷ 3 =, and the result is 120.

Formula

The combination formula counts the number of ways to choose r items from a set of n distinct items, where order does not matter. It is written as:

nCr = n! / [ r! × (n − r)! ]

• n – total number of items (must be a non‑negative integer)
• r – number of items chosen (must be between 0 and n)
• ! – factorial, meaning multiply all positive integers from 1 up to that number (e.g., 5! = 5×4×3×2×1 = 120)

The result is always an integer. For example, 5C2 = 10, because there are 10 different pairs you can pick from 5 items.

Worked Example

Using the default values from the calculator above (n = 10, r = 3):

1. Write the formula: 10C3 = 10! / [3! × (10−3)!]
2. Simplify the denominator: (10−3)! = 7!
3. So 10C3 = 10! / (3! × 7!)
4. Expand 10! / 7! = 10 × 9 × 8 = 720
5. Divide by 3! = 6: 720 / 6 = 120

Therefore, 10C3 = 120. The calculator above gives the same result.

When to Use nCr vs. nPr

The key question is whether order matters. Use combinations (nCr) when the arrangement is irrelevant, and permutations (nPr) when the order makes a difference.

• Combination example: Choosing 3 students from a class of 10 to form a committee – the committee is the same regardless of the order you pick them.
• Permutation example: Choosing 3 students from 10 to be president, vice‑president, and treasurer – each different assignment is a distinct outcome.

On most calculators, the nPr function is just above the nCr key (often accessed with the same SHIFT sequence). The permutation formula is nPr = n! / (n−r)!.

How to Find the nCr Button on Different Calculators

The exact key sequence varies slightly by model, but the pattern is consistent:

• Casio (fx‑82, fx‑991, etc.): Press SHIFT then the ÷ key (the nCr label is printed above it).
• Texas Instruments (TI‑30X, TI‑36X): Press the number n, then PRB (or MATH → PRB), scroll to nCr, press ENTER, then enter r.
• Sharp (EL‑531, etc.): Press 2ndF then the ÷ key.
• HP (RPN models): Enter n, press ENTER, enter r, then press nCr (often labeled directly on a key).

If your calculator does not have an nCr key, you can compute it manually using the factorial button (!) and the formula above, but be careful – factorials grow extremely fast, and intermediate values can overflow the display even when the final answer is small.

Common Mistakes and How to Avoid Them

• Entering n and r in the wrong order: nCr is symmetric (10C3 = 10C7), but only if r ≤ n. If you swap n and r, you may get an error or a wrong answer. Always enter the total first.
• Using nPr instead of nCr: For 10 items taken 3 at a time, nPr = 720, while nCr = 120. Check whether order matters before choosing the function.
• Forgetting that r can be 0 or n: nC0 = 1 and nCn = 1 for any n. This is correct – there is exactly one way to choose nothing or everything.
• Using non‑integers: n and r must be whole numbers. If you enter a decimal, the calculator may give an error or an undefined result.
• Confusing the factorial notation: The exclamation mark on the calculator is a function, not a key you press after the number. On most models, you press the number, then the ! key (often SHIFT ×).

Why Combinations Matter in Real Life

Combinations appear in probability, statistics, and everyday decision‑making. For example:

• Lottery odds: The number of possible ticket combinations is nCr. For a 6‑number lottery from 1–49, the total is 49C6 = 13,983,816.
• Card games: The number of possible 5‑card poker hands from a 52‑card deck is 52C5 = 2,598,960.
• Team selection: How many different 5‑person teams can be formed from 12 players? That is 12C5 = 792.
• Probability calculations: If you want the chance of drawing exactly 2 red balls from a bag, you often use combinations to count favourable outcomes.

In statistics, the binomial coefficient (another name for nCr) is the basis of the binomial distribution, which models the number of successes in a fixed number of independent trials.

Limits of the Calculator Function

Scientific calculators handle nCr well for moderate values, but there are practical limits:

• Large n: Even if the final answer fits in the display, intermediate factorial calculations can overflow. For example, 100C50 is about 1.0 × 10^29, but 100! is about 9.3 × 10^157 – most calculators can handle that, but some older models cannot.
• Memory: Some calculators store nCr results as floating‑point approximations, which may lose precision for very large numbers. For exact integer results, use a computer algebra system or an online calculator.
• Non‑integer inputs: The function is only defined for non‑negative integers. If you need a generalised version (for real numbers), you would use the gamma function, which is beyond the scope of a typical scientific calculator.

For most practical problems – choosing committees, lottery odds, card probabilities – the nCr button is perfectly accurate and fast.

Frequently Asked Questions

What does nCr mean on a calculator?

nCr stands for “n choose r� – the number of combinations of n items taken r at a time, where order does not matter. It is the same as the binomial coefficient.

How do I do nCr on a Casio calculator?

On most Casio models, press SHIFT then the ÷ key (the nCr label is printed above it). Enter n, press that key sequence, enter r, then press equals.

Is nCr the same as C(n, r)?

Yes, they are different notations for the same thing. C(n, r), nCr, and the binomial coefficient all mean the number of ways to choose r items from n without regard to order.

Why does my calculator give an error when I try nCr?

Most likely because r is greater than n, or one of the values is not a non‑negative integer. Check that you entered the total first and the chosen number second, and that both are whole numbers with r ≤ n.

Can I use nCr for probability problems?

Yes. To find the probability of a specific combination, divide the number of favourable combinations by the total number of possible combinations. For example, the probability of drawing a specific 5‑card poker hand is 1 / 52C5.

What is the difference between nCr and nPr on a calculator?

nCr counts combinations (order does not matter), while nPr counts permutations (order matters). For the same n and r, nPr is always larger because each combination can be arranged in r! different orders.

Related

• Age Calculator Image: What It Shows and How to Read It
• IGCSE Computer Science File Size Calculation Explained
• Calculate Screen Size From Height and Width

in n | Total items (n) | - | 10 | 0
in r | Chosen items (r) | - | 3 | 0
out comb | Combinations (nCr) | - | ncr(n, r) | 0