### Video Transcript

Given that π choose four is equal to five over two π, determine π choose three.

Letβs begin by recalling what we actually mean by this notation. π choose π is the number of ways of choosing our unique items from a total of π in the collection, assuming that order doesnβt matter. Itβs a combination. Itβs π factorial over π factorial times π minus π factorial. Initially, weβre told that π choose four is equal to five over two π. So letβs find an expression for π choose four. By letting π be equal to four, we find itβs equal to π factorial over four factorial times π minus four factorial. But of course this is equal to five over two π. So letβs form an equation in π. We get five over two π equals π factorial over four factorial times π minus four factorial.

We can simplify this a little bit. We know that π factorial is π times π minus one times π minus two and so on. So we can rewrite it as π times π minus one times π minus two times π minus three times π minus four factorial. And then we see that weβre going to be able to cancel a factor of π minus four factorial from our numerator and denominator, leaving us with the equation five over two π equals π times π minus one times π minus two times π minus three over four factorial. Letβs multiply both sides of our equation by four factorial, where four factorial is four times three times two times one, which is 24.

That means we get five over two times 24 which is 60π on the left-hand side. And on the right, we get π times π minus one times π minus two times π minus three. We subtract 60π from both sides, and weβre going to distribute our parentheses. By distributing π times π minus one times π minus two times π minus three, we get π to the fourth power minus six π cubed plus 11π squared minus six π. Letβs simplify by subtracting that 60π and the right-hand side of our equation now becomes π to the fourth power minus six π cubed plus 11π squared minus 66π.

Now, to solve this equation for π, weβre going to need to factor. So we spot that we can begin by factoring π from the right-hand side to get π times π cubed minus six π squared plus 11π minus 66. But how do we factor π cubed minus six π squared plus 11π minus 66? Well, we begin by recalling the factor theorem. This says that if π minus π is a factor of the function π of π, then π of π must be equal to zero. We can take a little guess as to what π might be by looking for factors of negative 66. Letβs try π is equal to six. π of six is six cubed minus six times six squared minus 11 times six minus 66 which is equal to zero.

So π of six is equal to zero, meaning π minus six must be a factor of our cubic. This means we can perform polynomial long division to fully factor the cubic. π cubed divided by π is π squared. Then, if we multiply π squared by both parts of our binomial, we get π cubed minus six π squared. Subtracting these terms and we get zero. And now, we bring down the next two terms. Thatβs 11π minus 66. Weβre going to divide 11π by π to give us 11. And now, we multiply 11 by both terms in our binomial.

That gives us 11π minus 66. And subtracting, we see we get a remainder of zero, which we expected since π minus six is a factor of our cubic. And so when we factor our equation, we get zero equals π time π minus six times π squared plus 11. One of the solutions to this equation is π equals zero. Another solution is when π minus six is equal to zero, so π is six. The equation π squared plus 11 equals zero, however, has no real solutions because if weβre trying to solve it, weβd be finding the square root of negative 11. We know that in π choose π, π must be a positive integer. So we choose the value π equal six and disregard any further solutions.

Letβs clear some space and determine π choose three if π is equal to six. Weβre going to evaluate six choose three. And so letting π be equal to six and π be equal to three in our earlier formula, we get six factorial over three factorial times three factorial. We write six factorial as six times five times four times three factorial. And then we cancel three factorial. But three factorial is six, so we can cancel another three factorial. And we find six choose three is simply equal to five times four which is 20. And so given that π choose four is five over two π, we find π choose three is equal to 20.