How to Calculate Binomial Coefficients

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How to Calculate Binomial Coefficients
Calculators are helpful when solving binomials because they solve simple arithmetic quickly. (calculator image by Szymon Apanowicz from Fotolia.com)

You have seven kinds of chocolate and you want to know how many combinations of three kinds can you make. Binomial coefficients are the mathematical way of picking k-item subsets from n original items. The common symbol for a binomial coefficient looks like a fraction inside parentheses without the fraction bar (n k), or nCk, and is read "n choose k." N choose k gives the number of k-subsets that are possible out of n different objects or choices. Use the following formula: nCk = n!/[(n-k)!*k!].

Skill level:
Moderate

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Things you need

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Instructions

  1. 1

    Set up an example, such as 5C3. Write 5C3 out explicitly in factorial form.

    From the problem, you have n = 5 and k = 3.

    The factorial form is 5!/[(5-3)!*3!].

  2. 2

    Solve what's in the parentheses first: (5-3) = 2, so the problem now reads: 5!/[2!*3!].

  3. 3

    Calculate n! (that is, n factorial), where n! = n(n-1)(n-2)...2*1.

    In the example problem: n=5, so n! = 5! = 54321 = 20321 = 6021 = 1201 = 120. Now the problem reads: 120/[2!*3!].

  4. 4

    Calculate (n-k)! In the example problem, (n-k)! = 2! from Step 2.

    2! = 21 = 2. Now the problem reads: 120/[23!].

  5. 5

    Calculate k! In the example problem, k! = 3! = 321 = 6*1 = 6.

    Now the problem reads: 120/[2*6].

  6. 6

    Solve the problem inside the brackets. In the example problem, [2*6] = 12

    Now the problem reads: 120/12.

  7. 7

    Simplify the fraction using division. In the example problem, 120/12 = 10

Tips and warnings

  • To solve binomial coefficients, you must be familiar with factorials. Do not attempt this without prior knowledge of factorials. For example: 5! means "5 factorial" and 5! = 5*4*3*2*1. If x is any positive integer, then x! = x*(x-1)*(x-2)*...*3*2*1.

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