Video: Simplifying Expressions Involving Factorials

Simplify (𝑛 + 2)!/𝑛!.

01:43

Video Transcript

Simplify 𝑛 plus two factorial divided by 𝑛 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. We need to multiply all of the integers or whole numbers from our number down to one. In order to calculate 𝑛 plus two factorial, we would need to multiply 𝑛 plus two by 𝑛 plus one by 𝑛 by 𝑛 minus one by 𝑛 minus two all the way down to one.

We notice that from 𝑛 onwards is equal to 𝑛 factorial. 𝑛 plus two factorial is therefore equal to 𝑛 plus two multiplied by 𝑛 plus one multiplied by 𝑛 factorial. If we substitute this into our question, we have 𝑛 plus two multiplied by 𝑛 plus one multiplied by 𝑛 factorial all divided by 𝑛 factorial. Any number divided by itself is equal to one. Therefore, 𝑛 factorial divided by 𝑛 factorial is one. We are therefore left with 𝑛 plus two multiplied by 𝑛 plus one.

Whilst we could distribute the parentheses or expand the brackets here, we don’t need to, as the expression is already in its simplest form. The simplified version of 𝑛 plus two factorial divided by 𝑛 factorial is 𝑛 plus two multiplied by 𝑛 plus one.

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