Lesson Video: Factorials | Nagwa Lesson Video: Factorials | Nagwa

# Lesson Video: Factorials Mathematics

In this video, we will learn how to find the factorial of any number 𝑛, which is the product of all integers less than or equal to n and greater than or equal to one, and we will learn how to find factorials to solve problems.

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### Video Transcript

In this video, we will learn how to find the factorial of any positive integer 𝑛 which is the product of all integers less than or equal to 𝑛 and greater than or equal to one. We will then consider how we can use factorials to solve problems. We will begin by looking at the definition of a factorial.

The factorial of a positive integer 𝑛 is the product of all the positive integers less than or equal to 𝑛. We use either of the notations shown, which are both read as 𝑛 factorial. And this is equal to 𝑛 multiplied by 𝑛 minus one multiplied by 𝑛 minus two and so on multiplied by two multiplied by one.

We also define the factorial of zero to be equal to one. That is, zero factorial is equal to one. For any integer 𝑛 such that 𝑛 is greater than or equal to one, then 𝑛 factorial is also equal to 𝑛 multiplied by 𝑛 minus one factorial. We can see this from the general definition, as 𝑛 minus one multiplied by 𝑛 minus two and so on multiplied by two multiplied by one is equal to 𝑛 minus one factorial. It is this property that is particularly important when solving complex problems involving factorials. However, in our first example, we will practice computing a factorial.

Evaluate four factorial.

We begin by recalling that the factorial of any positive integer 𝑛 is the product of all the positive integers less than or equal to 𝑛. This means that 𝑛 factorial is equal to 𝑛 multiplied by 𝑛 minus one multiplied by 𝑛 minus two and so on multiplied by two multiplied by one. Four factorial is therefore equal to four multiplied by three multiplied by two multiplied by one.

As multiplication is commutative, we can multiply our integers in any order. For example, four multiplied by three is equal to 12. Two multiplied by one is equal to two. And then multiplying 12 by two gives us an answer of 24. Alternatively, we could multiply four by three, giving us 12, then multiply this by two, giving us 24, and finally multiplying 24 by one gives us a final answer of 24. This confirms that four factorial is equal to 24.

Our next example is a more complicated problem involving factorials.

Simplify the expression six factorial over four factorial minus 27 factorial over 28 factorial. Give you answer as a fraction.

We recall that since 𝑛 factorial is the product of all the positive integers less than or equal to 𝑛, then 𝑛 factorial is equal to 𝑛 multiplied by 𝑛 minus one multiplied by 𝑛 minus two and so on all the way down to one. This leads us to a key property of factorials. 𝑛 factorial is equal to 𝑛 multiplied by 𝑛 minus one factorial. This means that instead of calculating each of the factorials individually, we can simplify and look for common factors.

We can write six factorial as six multiplied by five multiplied by four factorial. This means that the first term in our expression can be written as six multiplied by five multiplied by four factorial divided by four factorial. We can then cancel a factor of four factorial from the numerator and denominator, leaving us with six multiplied by five.

In the same way, we can write the denominator of our second term as 28 multiplied by 27 factorial. This time, we can cancel the factor of 27 factorial on the numerator and denominator. And this term simplifies to one over 28. Six multiplied by five is equal to 30. So we need to subtract one twenty-eighth from 30. We could write this as a mixed number as 29 and twenty-seven twenty-eighth. However, as we want to give our answer just as a fraction, we will convert the number 30 or 30 over one to a fraction over 28.

In order to do this, we will multiply 30 by 28. As 28 multiplied by three is 84, 28 multiplied by 30 is 840. This means that 30 or 30 over one is equal to 840 over 28. As our denominators are the same, we simply subtract the numerators. And we can therefore conclude that six factorial over four factorial minus 27 factorial over 28 factorial written as a fraction is 839 over 28.

In our final example, we will use our knowledge of factorials to solve an algebraic equation.

Find the solution set of one over 𝑛 plus seven factorial plus one over 𝑛 plus eight factorial is equal to 256 over 𝑛 plus nine factorial.

In order to answer this question, we begin by recalling that for any integer 𝑛 greater than or equal to one, then 𝑛 factorial is equal to 𝑛 multiplied by 𝑛 minus one factorial. The fact that 𝑛 must be greater than or equal to one will be important when finding the solution set.

When dealing with a problem where the sum of two fractions is equal to another fraction, it is often useful to try and eliminate the denominators first. In this question, we will multiply all three terms by 𝑛 plus nine factorial. This gives us 𝑛 plus nine factorial over 𝑛 plus seven factorial plus 𝑛 plus nine factorial over 𝑛 plus eight factorial is equal to 256 multiplied by 𝑛 plus nine factorial over 𝑛 plus nine factorial. On the right-hand side of our equation, we can cancel the common factor of 𝑛 plus nine factorial, leaving us with 256.

Using the fact that 𝑛 factorial is equal to 𝑛 multiplied by 𝑛 minus one factorial and it is also equal to 𝑛 multiplied by 𝑛 minus one multiplied by 𝑛 minus two factorial, we can rewrite the left-hand side of our equation as shown. In the first term, we can cancel a factor of 𝑛 plus seven factorial. And in the second term, we cancel a factor of 𝑛 plus eight factorial. We now have an equation that no longer contains any fractions. 𝑛 plus nine multiplied by 𝑛 plus eight plus 𝑛 plus nine is equal to 256.

We can simplify the left-hand side either by taking out a factor of 𝑛 plus nine or by using the FOIL method to distribute our parentheses. 𝑛 plus nine multiplied by 𝑛 plus eight is equal to 𝑛 squared plus eight 𝑛 plus nine 𝑛 plus 72. And when we add 𝑛 plus nine to this, we get an answer of 256. We can then group or collect the like terms on the left-hand side. This gives us 𝑛 squared plus 18𝑛. And when we subtract 256 from both sides, we have negative 175. We now have a quadratic equation 𝑛 squared plus 18𝑛 minus 175 is equal to zero.

Our next step is to factor the expression on the left-hand side into two sets of parentheses. Since the coefficient of 𝑛 squared is equal to one, we know that the first term in each of them will be 𝑛. And the second terms will have a sum of positive 18 and a product of negative 175. One factor pair of 175 is 25 and seven. This means that multiplying 25 by negative seven gives us negative 175. And since 25 plus negative seven is 18, our two sets of parentheses are 𝑛 plus 25 and 𝑛 minus seven.

As the product of 𝑛 plus 25 and 𝑛 minus seven equals zero, then either 𝑛 plus 25 equals zero or 𝑛 minus seven equals zero. This gives us two possible solutions 𝑛 equals negative 25 and 𝑛 equals seven. As already mentioned, we know that 𝑛 must be greater than or equal to one, as factorials are only defined for nonnegative integers. The value of 𝑛 that satisfies the equation is therefore equal to seven. And the solution set of the equation one over 𝑛 plus seven factorial plus one over 𝑛 plus eight factorial is equal to 256 over 𝑛 plus nine factorial contains the number seven.

We will now summarize the key points from this video.

The factorial of a positive integer 𝑛 is defined as the product of all positive integers less than or equal to 𝑛 such that 𝑛 factorial is as shown. The key property of the factorial is that 𝑛 factorial is equal to 𝑛 multiplied by 𝑛 minus one factorial. And we can use this to simplify expressions involving factorials. Whilst it was not covered by a specific example in this video, when trying to find an unknown integer given its factorial, we divide by consecutive positive integers.

This means that in order to find the value of 𝑛 such that 𝑛 factorial is equal to 120, we divide by the integers one, two, three, and so on. 120 divided by one is 120. Dividing this by two gives us 60. Dividing 60 by three gives us 20. 20 divided by four is equal to five. And finally, five divided by five is equal to one. We can therefore conclude that 120 is equal to five multiplied by four multiplied by three multiplied by two multiplied by one, which can be written as five factorial. The value of 𝑛 such that 𝑛 factorial is 120 is five.