Worksheet on Combinations and Permutations

The formula for combinations can be written by using three factorials.
Combination Formula. C.K.Taylor

Permutations and combinations are two concepts that related to ideas in probability.  These two topics are very similar and are easy to get confused. In both cases we start with a set containing a a total of n elements.  Then we count r of these elements. The way in which we count these elements determines if we are working with a combination or with a permutation.

Ordering and Arrangement

The key things to remember when distinguishing between combinations and permutations has to do with order and arrangements.  Permutations deal with situations when the order that we choose the objects is important.  We can also think of this as being equivalent to the idea of arranging objects

In combinations we are not concerned with what order we selected our objects. We only need this concept, and the formulas for combinations and permutations to solve problems dealing with this topic.

Practice Problems

To get good at something, it takes some practice.  Here are some practice problems with solutions to help you to straighten out the ideas of permutations and combinations. A version with answers is here.  After starting with just basic calculations, you can use what you know to determine if a combination or permutation is being referred to.

  1. Use the formula for permutations to calculate P( 5, 2 ).
  2. Use the formula for combinations to calculate C( 5, 2 ).
  3. Use the formula for permutations to calculate P( 6, 6 ).
  4. Use the formula for combinations to calculate C( 6, 6 ).
  5. Use the formula for permutations to calculate P( 100, 97 ).
  6. Use the formula for combinations to calculate C( 100, 97 ).
  7. It’s election time at a high school that has a total of 50 students in the junior class. How many ways can a class president, class vice president, class treasurer,and class secretary be chosen if each student may only hold one office?
  8. The same class of 50 students wants to form a prom committee. How many ways can a four person prom committee be selected from the junior class?
  9. If we want to form a group of five students and we have 20 to choose from, how many ways is this possible?
  10. How many ways can we arrange four letters from the word “computer” if repetitions are not allowed, and different orders of the same letters count as different arrangements?
  11. How many ways can we arrange four letters from the word “computer” if repetitions are not allowed, and different orders of the same letters count as the same arrangement?
  1. How many different four digit numbers are possible if we can choose any digits from 0 to 9 and all of the digits must be different?
  2. If we are given a box containing seven books, how many ways can we arrange three of them on a shelf?
  3. If we are given a box containing seven books, how many ways can we choose collections of three of them from the box?