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N Choose R on Graphing Calculator

Reviewed by Calculator Editorial Team

Calculating n choose r (also known as combinations) is a fundamental concept in combinatorics. This guide explains how to perform these calculations on a graphing calculator, including step-by-step instructions, formulas, and practical examples.

What is n choose r?

In combinatorics, "n choose r" refers to the number of ways to choose r items from a set of n distinct items without regard to the order of selection. This is often written as C(n, r) or nCr.

The calculation is based on the combination formula:

C(n, r) = n! / (r! × (n - r)!)

Where "!" denotes factorial, which is the product of all positive integers up to that number.

How to calculate n choose r

To calculate n choose r manually or on a calculator, follow these steps:

  1. Determine the values of n and r
  2. Calculate the factorial of n (n!)
  3. Calculate the factorial of r (r!)
  4. Calculate the factorial of (n - r) ((n - r)!)
  5. Multiply r! and (n - r)! together
  6. Divide n! by the product from step 5

Note: Factorials grow very quickly, so calculations for large n and r may require a calculator or software.

Using a graphing calculator

Most graphing calculators have built-in functions for combinations. Here's how to use them:

  1. Enter the values for n and r
  2. Use the combination function (often labeled as nCr or C(n, r))
  3. Press the equals key to get the result

For calculators without a built-in combination function, you can use the factorial function:

nCr = factorial(n) / (factorial(r) × factorial(n - r))

Some calculators may use different notation, so consult your manual if needed.

Example calculations

Let's calculate C(5, 2):

  1. n = 5, r = 2
  2. 5! = 120
  3. 2! = 2
  4. (5 - 2)! = 3! = 6
  5. 2! × 3! = 2 × 6 = 12
  6. 120 / 12 = 10

So, C(5, 2) = 10, meaning there are 10 ways to choose 2 items from a set of 5.

Common mistakes

When calculating n choose r, be aware of these common errors:

  • Confusing combinations with permutations (order matters in permutations)
  • Using the wrong factorial values
  • Forgetting to subtract r from n when calculating (n - r)!
  • Not simplifying the calculation when possible

Tip: Always double-check your calculations, especially with larger numbers.

FAQ

What is the difference between combinations and permutations?

Combinations (n choose r) count the number of ways to choose r items without regard to order. Permutations count the number of ways to arrange r items where order matters.

Can I calculate n choose r for large numbers?

Yes, but be aware that factorials grow very quickly. For very large numbers, you may need specialized software or programming.

What if r is greater than n?

By definition, n choose r is 0 when r > n because you can't choose more items than are available.

Is n choose r the same as Pascal's triangle?

Yes, the values in Pascal's triangle correspond to combinations. The nth row represents C(n, r) for r from 0 to n.