Evaluate six factorial divided by five factorial.
We recall that 𝑛 factorial is equal to 𝑛 multiplied by 𝑛 minus one multiplied by 𝑛 minus two all the way down to one. To calculate the factorial of 𝑛, we multiply all of the integers from 𝑛 down to one. This means that six factorial is equal to six multiplied by five multiplied by four multiplied by three multiplied by two multiplied by one. Five factorial is equal to five multiplied by four multiplied by three multiplied by two multiplied by one.
We notice that this is part of six factorial. Six factorial can, therefore, be rewritten as six multiplied by five factorial. We need to divide this by five factorial. As we have five factorial on the numerator and denominator, these will cancel as any number divided by itself gives us one. Six factorial divided by five factorial is, therefore, equal to six.
To solve any problem of this type, it is quicker to use the rule that 𝑛 factorial is equal to 𝑛 multiplied by 𝑛 minus one factorial. If we didn’t recognize this, we could’ve calculated six factorial by multiplying the integers from six to one. This gives us an answer of 720. Five factorial is equal to 120. When we divide 720 by 120, we once again get an answer of six.