Question Video: Counting Using Permutations Mathematics • Second Year of Secondary School

Video Transcript

Evaluate eight P five.

The notation in this question tells us that we are dealing with permutations. We know that a permutation is an arrangement of a collection of items. When dealing with permutations, order matters and repetition is not allowed. We can calculate the number of permutations using the formula 𝑛 factorial divided by 𝑛 minus 𝑟 factorial. This is the number of ways we can select 𝑟 elements from a collection of 𝑛 elements.

In this question, we need to calculate the number of ways we can select five elements from a group of eight elements without repetition and where order matters. Substituting in our values, we have eight factorial divided by eight minus five factorial. As eight minus five is equal to three, this simplifies to eight factorial divided by three factorial. We recall that 𝑛 factorial is equal to 𝑛 multiplied by 𝑛 minus one factorial. This means that we can rewrite the numerator as eight multiplied by seven multiplied by six multiplied by five multiplied by four multiplied by three factorial.

We can then divide the numerator and denominator by three factorial. Multiplying the five integers from eight to four inclusive gives us 6720. There are 6720 ways we can select five items from a group of eight items. It is important to note that there are different notations that can be used in these type of questions, some of which are shown.

We could also have used a scientific calculator to work out the answer. We simply type the 𝑛 value followed by the 𝑛P𝑟 button and then the 𝑟 value. Pressing “equals” will then give us our answer. In this case, eight P five is equal to 6720.