Video Transcript
If 𝑥 minus 47 factorial times
𝑥𝑃47 equals 3906 times 𝑥 minus two factorial, find the value of 𝑥.
To solve for 𝑥, we’ll need to
understand what is meant by 𝑥𝑃47 and also what is meant by factorial. The notation 𝑛𝑃𝑟, where 𝑛 and
𝑟 are both nonnegative integers, is defined as the fraction 𝑛 factorial divided by
𝑛 minus 𝑟 factorial. It turns out that this calculation
gives the number of ways to select 𝑟 unique objects from a group of 𝑛 unique
objects if order matters. In other words, it’s the number of
permutations of 𝑟 things taken from a group of 𝑛 things, hence the notation 𝑃 for
permutation.
The formula for 𝑛𝑃𝑟 also uses
the factorial notation, so let’s define factorial. For a positive integer 𝑛, 𝑛
factorial is defined as the product of all of the positive integers from one to 𝑛
inclusive. So 𝑛 factorial is 𝑛 times 𝑛
minus one times 𝑛 minus two etcetera times three times two times one. As a side note, we also say that
zero factorial is equal to one.
Anyway, looking at the definition
of factorial, we see that on the right-hand side we have 𝑛 times the product of all
of the positive integers from one to 𝑛 minus one, in other words, 𝑛 times 𝑛 minus
one factorial. This actually gives an alternate
way to define factorial. 𝑛 factorial is 𝑛 times 𝑛 minus
one factorial. Anyway, these three definitions are
all that we need to solve our equation. Let’s start with the right-hand
side, and we’ll expand out 𝑥𝑃47 in terms of factorials.
So we have that 𝑥 minus 47
factorial times 𝑥𝑃47 is equal to 𝑥 minus 47 factorial times 𝑥 factorial divided
by 𝑥 minus 47 factorial. To arrive at this expression, we
replaced 𝑛 with 𝑥 and 𝑟 with 47 in the definition of 𝑛𝑃𝑟. Looking at the expression on the
right-hand side, we see we have 𝑥 minus 47 factorial divided by 𝑥 minus 47
factorial. But anything divided by itself is
just one, so we’re left with 𝑥 factorial on the right-hand side. But we know that this expression is
equal to the right-hand side of our original equation. So we can now set up the equation
𝑥 factorial equals 3906 times 𝑥 minus two factorial.
To continue simplifying our
expression, let’s see what happens when we use our second definition for factorial
twice. If we let 𝑛 minus one take the
place of 𝑛, we have that 𝑛 minus one factorial is equal to 𝑛 minus one times 𝑛
minus two factorial. But if we substitute this back into
our original definition for 𝑛 factorial, we see that 𝑛 factorial is 𝑛 times 𝑛
minus one times 𝑛 minus two factorial. In fact, by repeatedly using this
definition, we can expand 𝑛 factorial for as many terms as we need. For our equation, twice will be
enough because we have an 𝑥 minus two factorial on the right-hand side and an 𝑥
factorial on the left-hand side.
So expanding 𝑥 factorial, we have
𝑥 times 𝑥 minus one times 𝑥 minus two factorial is equal to 3906 times 𝑥 minus
two factorial. Now we have a common factor of 𝑥
minus two factorial on both sides. So let’s divide both sides of this
equation by 𝑥 minus two factorial. This leaves us with 𝑥 times 𝑥
minus one equals 3906. This is a quadratic equation that
we can solve by whatever algebraic technique we prefer. Let’s illustrate one way that is
particularly useful for problems involving factorials because we know that 𝑥 must
be an integer.
We observe that if 𝑥 is restricted
to be an integer, 𝑥 and 𝑥 minus one are consecutive integers, which means that
their value is quite similar. This means that 𝑥 minus one is
approximately equal to 𝑥, so 𝑥 times 𝑥 minus one is approximately 𝑥 squared. Now this is not entirely accurate,
but it does mean that 𝑥 is pretty close to the square root of 3906. We know this can’t be perfectly
accurate because 𝑥 is an integer and the square root of 3906 is not an integer. However, once we calculate the
square root of 3906, we know that 𝑥 is one of the integers very near to that
number. And usually, we’ll only need to try
one or two values before we find the correct one.
The square root of 3906 is
approximately 62.5. We’ve only reported this number to
one decimal place because all we care about are the nearby integers. Our immediate possibilities for 𝑥
are then 63 and 62, which are the two integers nearest to 62.5. Of these two numbers, 63 is the
most sensible first guess because we know that 𝑥 times 𝑥 minus one is 3906 and 𝑥
is bigger than 𝑥 minus one. And indeed, we find that 63 times
62 is exactly 3906. This confirms our educated guess
that 𝑥 is in fact 63.
Approximating 𝑥 by taking a root
is a generally applicable technique whenever we have a product of several
consecutive positive integers. If we have 𝑛 consecutive integers,
we take the 𝑛th root of their product. And the value that we get is fairly
close to the average of those consecutive integers. Once we’ve approximated the average
value of the consecutive integers, a little trial and error gives us the full
list.
