# Question Video: Evaluating Factorials

Find 𝑧 such that 𝑧! = 120.

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### Video Transcript

Find 𝑧 such that 𝑧 factorial is equal to 120.

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 multiply all the whole numbers or integers from our number 𝑛 all the way 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. This is equal to six. We might notice, however, that this calculation contains two factorial. Therefore, three factorial is equal to three multiplied by two factorial.

This leads us to a general rule. 𝑛 factorial is equal to 𝑛 multiplied by 𝑛 minus one factorial. We can rewrite the calculation for four factorial four multiplied by three multiplied by two multiplied by one as four multiplied by three factorial. We need to multiply four by six, which gives us an answer of 24. In the same way, five factorial is equal to five multiplied by four factorial. We need to multiply five by 24. This is equal to 120. This was the value we wanted in our question. As five factorial is equal to 120, our value of 𝑧 is five.