How to Solve Permutations and Combinations? (+FREE Worksheet!)

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Step by step guide to solve Permutations and Combinations

• Permutations: The number of ways to choose a sample of \(k\) elements from a set of \(n\) distinct objects where order does matter, and replacements are not allowed. For a permutation problem, use this formula:
  \(\color{blue}{_{n}P_{k }= \frac{n!}{(n-k)!}}\)
• Combination: The number of ways to choose a sample of \(r\) elements from a set of \(n\) distinct objects where order does not matter, and replacements are not allowed. For a combination problem, use this formula:
  \(\color{blue}{_{ n}C_{r }= \frac{n!}{r! (n-r)!}}\)
• Factorials are products, indicated by an exclamation mark. For example, \(4!\) Equals: \(4×3×2×1\). Remember that \(0!\) is defined to be equal to \(1\).

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Permutations and Combinations – Example 1:

How many ways can the first and second place be awarded to \(10\) people?

Solution:

Since the order matters, we need to use the permutation formula where \(n\) is \(10\) and \(k\) is \(2\). Then: \(\frac{n!}{(n-k)!}=\frac{10!}{(10-2)!}=\frac{10!}{8!}=\frac{10×9×8!}{8!}\), remove \(8!\) from both sides of the fraction. Then: \(\frac{10×9×8!}{8!}=10×9=90\)

Permutations and Combinations – Example 2:

How many ways can we pick a team of \(3\) people from a group of \(8\)?

Solution:

Since the order doesn’t matter, we need to use a combination formula where \(n\) is \(8\) and \(r\) is \(3\). Then: \(\frac{n!}{r! (n-r)!}=\frac{8!}{3! (8-3)!}=\frac{8!}{3! (5)!}=\frac{8×7×6×5!}{3! (5)!}\), remove \(5!\) from both sides of the fraction. Then: \(\frac{8×7×6}{3×2×1}=\frac{336}{6}=56\)

Exercises for Solving Permutations and Combinations

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Calculate the value of each.

1. \(\color{blue}{4!=}\)
2. \(\color{blue}{4!×3!=}\)
3. \(\color{blue}{5!=}\)
4. \(\color{blue}{6!+3!=}\)
5. There are \(7\) horses in a race. In how many different orders can the horses finish?
6. In how many ways can \(6\) people be arranged in a row?

Download Combinations and Permutations Worksheet

1. \(\color{blue}{24}\)
2. \(\color{blue}{144}\)
3. \(\color{blue}{120}\)
4. \(\color{blue}{726}\)
5. \(\color{blue}{5,040}\)
6. \(\color{blue}{720}\)

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