Video: Factorials

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 number 𝑛, which is the product of all integers less than or equal to 𝑛 and greater than or equal to one. We will also learn how to find factorials to solve problems and solve problems containing permutations and factorials. Let’s begin by looking at a written and algebraic 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 the notation 𝑛 followed by an exclamation mark, which is read as 𝑛 factorial. 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. We also define the factorial of zero to be equal to one; that is, zero factorial equals one. We also know that for any integer 𝑛 greater than or equal to one, 𝑛 factorial is equal to 𝑛 multiplied by 𝑛 minus one factorial. We can see that from the general rule for 𝑛 factorial above. This property will be really useful when solving more complicated problems in this video. However, we will begin by solving a straightforward problem.

Evaluate four factorial.

We recall 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 all the way down to one. Four factorial is therefore equal to four multiplied by three multiplied by two multiplied by one. Four multiplied by three is equal to 12. Multiplying this by two gives us 24, and multiplying 24 by one is also 24. We can multiply the integers four, three, two, and one in any order to give us an answer of 24. Therefore, four factorial equals 24.

In our next question, we will solve a more complicated problem.

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

We recall that 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 means that we could calculate six factorial by multiplying six by five, by four, by three, by two, and by one. Whilst this wouldn’t be too difficult for the first fraction, working out 27 factorial and 28 factorial in this way would be very time-consuming. We can therefore recall another rule for calculating 𝑛 factorial. It is equal to 𝑛 multiplied by 𝑛 minus one factorial. We could also see that 𝑛 factorial is equal to 𝑛 multiplied by 𝑛 minus one multiplied by 𝑛 minus two factorial. This allows us to rewrite six factorial as six multiplied by five multiplied by four factorial.

The first term in our question simplifies to six multiplied by five multiplied by four factorial all divided by four factorial. As the four factorials cancel, we are left with six multiplied by five. This is equal to 30. We can use this method again for the second fraction as 28 factorial is equal to 28 multiplied by 27 factorial. This time, the 27 factorials cancel, leaving us with one over 28. We need to subtract one over 28 or one twenty-eighth from 30. This is equal to the mixed number 29 and twenty-seven twenty-eighths.

In order to write our answer just as a fraction, we will need to convert this into an improper or top-heavy fraction. We do that by first multiplying the whole number 29 by the denominator 28. This is equal to 812. We then add the numerator of 27, giving us 839. The expression six factorial over four factorial minus 27 factorial over 28 factorial is equal to the fraction 839 over 28.

In our next question, we’ll 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.

There are lots of ways of starting this question. One way would be to multiply both sides by 𝑛 plus nine factorial. Multiplying the first term by 𝑛 plus nine factorial gives us 𝑛 plus nine factorial over 𝑛 plus seven factorial. The second term on the left-hand side becomes 𝑛 plus nine factorial over 𝑛 plus eight factorial. As 𝑛 plus nine factorial divided by 𝑛 plus nine factorial is equal to one, the right-hand side becomes 256.

We recall that π‘Ÿ factorial is equal to π‘Ÿ multiplied by π‘Ÿ minus one factorial. This means that 𝑛 plus nine factorial can be rewritten as 𝑛 plus nine multiplied by 𝑛 plus eight factorial or 𝑛 plus nine multiplied by 𝑛 plus eight multiplied by 𝑛 plus seven factorial. The first term therefore simplifies to 𝑛 plus nine multiplied by 𝑛 plus eight. The second term simplifies to 𝑛 plus nine. 𝑛 plus nine multiplied by 𝑛 plus eight plus 𝑛 plus nine is equal to 256.

We can distribute the parentheses or expand the brackets using the FOIL method. Multiplying the first terms gives us 𝑛 squared, the outer terms eight 𝑛, the inner terms nine 𝑛, and the last terms 72. We now have an equation 𝑛 squared plus eight 𝑛 plus nine 𝑛 plus 72 plus 𝑛 plus nine is equal to 256. By collecting like terms, the left-hand side simplifies to 𝑛 squared plus 18𝑛 plus 81. We can then subtract 256 from both sides of the equation such that 𝑛 squared plus 18𝑛 minus 175 is equal to zero.

We can now factor this quadratic expression into two sets of parentheses. The first term in each of them is 𝑛, as 𝑛 multiplied by 𝑛 is 𝑛 squared. The second terms will have a sum of 18 and a product of negative 175. 25 multiplied by seven is 175. This means that positive 25 multiplied by negative seven is negative 175. The numbers positive 25 and negative seven also have a sum of 18. As this expression equals zero, one of our parentheses must be equal to zero. This means that either 𝑛 is equal to negative 25 or 𝑛 equals seven. Factorials are only defined for nonnegative integers. This means we can discard the solution 𝑛 equals negative 25. The value of 𝑛 that satisfies the equation is 𝑛 equals seven. The solution set of the equation just contains the number seven.

Our last question involves permutations and factorials. Before moving on to this, we will recall the definition of a permutation. A permutation is a rearrangement of a collection of items. It is defined as the number of ways we can order π‘Ÿ elements from a set of 𝑛 elements with no repetition. We write this as subscript 𝑛 capital P subscript π‘Ÿ. It is just read as 𝑛Pπ‘Ÿ. It is defined by 𝑛 factorial divided by 𝑛 minus π‘Ÿ factorial. For example, nine P five is equal to nine factorial divided by nine minus five factorial. This simplifies to nine factorial divided by four factorial.

Using the property that 𝑛 factorial is equal to 𝑛 multiplied by 𝑛 minus one factorial, nine factorial is equal to nine multiplied by eight multiplied by seven multiplied by six multiplied by five multiplied by four factorial. As the four factorial cancels, we can then multiply the five integers nine, eight, seven, six, and five, giving us 15,120.

We will now answer a question involving permutations and factorials.

Given that 𝑛Pπ‘Ÿ is equal to 504 and π‘Ÿ factorial equals six, find the values of 𝑛 and π‘Ÿ.

We recall that when dealing with permutations, 𝑛Pπ‘Ÿ is equal to 𝑛 factorial over 𝑛 minus π‘Ÿ factorial. We are also told in the question that π‘Ÿ factorial is equal to six. This is a factorial we can work out quite easily. We know that three multiplied by two multiplied by one is equal to six. This means that three factorial is equal to six. Our value of π‘Ÿ is therefore equal to three.

We were told that 𝑛Pπ‘Ÿ is equal to 504. Therefore, 𝑛P three equals 504. Substituting π‘Ÿ equals three into our general formula for permutations, we have 𝑛 factorial divided by 𝑛 minus three factorial is equal to 504. 𝑛 factorial can be rewritten as 𝑛 multiplied by 𝑛 minus one multiplied by 𝑛 minus two multiplied by 𝑛 minus three factorial. Dividing this by 𝑛 minus three factorial, we are left with 𝑛 multiplied by 𝑛 minus one multiplied by 𝑛 minus two. This is equal to 504. We are therefore looking for three consecutive integers that multiply to give us 504.

We could try to work out these values using trial and improvement. However, there is a trick we can use to find three consecutive integers that multiply to give a number. We begin by taking the cube root of that number. The cube root of 504 is 7.958 and so on. Well, how does this help us? This is not an integer. What we can do is take the integers on either side of this number. In this case, these are seven and eight. We can now divide our number, in this case 504, by the two integers. 504 divided by seven is equal to 72. Therefore, seven multiplied by 72 is 504. We now divide 72 by the second integer eight. 72 divided by eight is equal to nine. Therefore, eight multiplied by nine is 72.

We have now written 504 as the product of three consecutive integers. These integers are seven, eight, and nine, which correspond to 𝑛 minus two, 𝑛 minus one, and 𝑛, respectively. Our value of 𝑛 is nine. If 𝑛Pπ‘Ÿ equals 504 and π‘Ÿ factorial is six, π‘Ÿ equals three and 𝑛 equals nine. There are slightly different methods we could’ve used to calculate π‘Ÿ and then calculate 𝑛. Let’s firstly go back to the fact that we know that π‘Ÿ factorial is equal to six.

When trying to find an unknown integer given its factorial, we can divide by consecutive positive integers. This means that we begin by dividing our number six by one. This is equal to six. We then divide by the next positive integer two. Six divided by two is equal to three. We then divide by the next positive integer three, and three divided by three is equal to one. As six divided by one divided by two divided by three is equal to one, then six is also equal to three multiplied by two multiplied by one. We have once again proved that three factorial is equal to six. Therefore, π‘Ÿ equals three.

When we got to the stage that 𝑛 multiplied by 𝑛 minus one multiplied by 𝑛 minus two was equal to 504, we could’ve used our knowledge of prime factors to try and rearrange them into consecutive integers. 504 is equal to two multiplied by 252. 252 is equal to two multiplied by 126. Repeating this process by dividing by prime numbers, we can write 504 as a product of its prime factors. 504 is equal to two multiplied by two multiplied by two multiplied by seven multiplied by three multiplied by three. This can be rewritten as two cubed multiplied by seven multiplied by three squared. Two cubed is equal to eight, and three squared is equal to nine. Once again, we have three consecutive integers seven, eight, and nine such that seven is equal to 𝑛 minus two, eight is equal to 𝑛 minus one, and nine is equal to 𝑛.

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. The key property of the factorial is that 𝑛 factorial is equal to 𝑛 multiplied by 𝑛 minus one factorial. We can use this to simplify expressions using factorials and also solve factorial equations. When trying to find an unknown integer given its factorial, we divide by consecutive positive integers until we reach an answer of one. Finally, we saw that the number of permutations of size π‘Ÿ taken from a set of size 𝑛 is given by 𝑛Pπ‘Ÿ is equal to 𝑛 factorial divided by 𝑛 minus π‘Ÿ factorial.

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