# Question Video: Applications of the Counting Principle

Determine the number of ways of selecting 2 teachers from 19.

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### Video Transcript

Determine the number of ways of selecting two teachers from 19.

In order to answer this question, we need to use our knowledge of permutations. A permutation is an arrangement of a collection of items without repetition and where order matters. The notation for this is 𝑛P𝑟, where 𝑟 is the number of items we are selecting and 𝑛 is the total number of items. This can be calculated using the formula 𝑛 factorial divided by 𝑛 minus 𝑟 factorial.

In this question, we have a total of 19 teachers. Therefore, 𝑛 is equal to 19. We need to select two of them. Therefore, 𝑟 is equal to two. 19P two is therefore equal to 19 factorial divided by 19 minus two factorial. The denominator here simplifies to 17 factorial. We recall at this stage that 𝑛 factorial is equal to 𝑛 multiplied by 𝑛 minus one factorial. This means that we can rewrite 19 factorial as 19 multiplied by 18 multiplied by 17 factorial. Dividing the numerator and denominator by 17 factorial leaves us with 19 multiplied by 18. This is equal to 342. There are 342 ways of selecting two teachers from 19.

We could also have worked out the answer using a scientific calculator. We type in 19 followed by the 𝑛P𝑟 button and then two. Pressing equals would give us an answer of 342.