Find the value of 𝑛 such that 𝑛 factorial is equal to 720.
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. We must multiply all the integers from our number 𝑛 down to one. This means that two factorial is equal to two multiplied by one. This is equal to two. Three factorial is equal to three multiplied by two multiplied by one.
There is an easier way to calculate this, though, as we know that two multiplied by one is two factorial. As two factorial is equal to two and three multiplied by two is six, then three factorial equals six. Four factorial is equal to four multiplied by three multiplied by two multiplied by one. This is the same as four multiplied by three factorial. Three factorial was equal to six. Therefore, four factorial is equal to four multiplied by six, which is 24.
This leads us to a general rule that 𝑛 factorial is equal to 𝑛 multiplied by 𝑛 minus one factorial. Five factorial is, therefore, equal to five multiplied by four factorial. As four factorial was 24, we need to multiply five by 24. This gives us an answer of 120. We are looking for the factorial that gives us an answer of 720. So this is still too small.
Using our rule six factorial is equal to six multiplied by five factorial, we need to multiply 120 by six. As six multiplied by 100 is 600 and six multiplied by 20 is 120, six multiplied by 120 is equal to 720. As six factorial equals 720, our value of 𝑛 is six.