### Video Transcript

Evaluate seven permute four divided
by five permute two.

So to evaluate this, we need to
work out how we actually find out the value of something like seven permute
four. Well, we actually have a general
formula that tells us that 𝑛 permute 𝑟 is equal to 𝑛 factorial over 𝑛 minus 𝑟
factorial. So with this formula, it’s worth
reminding yourself what factorial is.

And the factorial shown here — we
have five factorial — is five multiplied by four multiplied by three multiplied by
two multiplied by one. And to find any factorial, all we
do is actually multiply all the positive integers that are less than or equal to the
number which is in the factorial.

Okay, so now we know this. Let’s evaluate our seven permute
four divided by five permute two. First of all, I’m actually gonna
find seven permute four. And using our general formula is
gonna be equal to seven factorial divided by seven minus four factorial which is
gonna be equal to seven factorial over three factorial.

So now, we’re just gonna show what
this means. It means we’d have seven multiplied
by six multiplied by five multiplied by four multiplied by three multiplied by two
multiplied by one cause that’s our seven factorial then divided by three multiplied by two
multiplied by one. Well, then, our three factorial
actually cancels out. So we can actually remove three
multiplied by two multiplied by one from the numerator and the denominator. So then, we’re left with seven
multiplied by six multiplied by five multiplied by four which gives us a value of
840.

So great, we’ve now found the value
of seven permute four. Let’s move on and find the value of
five permute two. Well, five permute two is equal to
five factorial over five minus two factorial. And again, we got this from our
general formula which is gonna be equal to five factorial over three factorial. So again, we just put in this extra
step to show you what’s happening. So we’ve got five multiplied by
four multiplied by three multiplied by two multiplied by one divided by three
multiplied by two multiplied by one. Again, our three factorial
cancels. So we’re just left with five
multiplied by four. So therefore, we have a value of
five permute two of 20.

Okay, great, so now, we can
actually move on to the final stage which is to evaluate our starting question. So therefore, we can say that seven
permute four over five permute two is gonna be equal to 840 divided by 20 which is
gonna give us a final answer of 42.