Video: Evaluating Permutations

Evaluate ⁷𝑃₄/⁵𝑃₂.

02:28

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.

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