Simplify the expression six factorial over four factorial minus 27 factorial over 28 factorial. Give your answer as a fraction.
So the first thing about this question that is key is that we need to understand a notation has been used. And in this question, we’re using something called factorial. Now, if we have 𝑛 factorial, and this is equal to 𝑛 multiplied by 𝑛 minus one multiplied by 𝑛 minus two, then it would be multiplied by 𝑛 minus three, etcetera. And this would carry on until you get two multiplied by one. For example, if we have four factorial, this will be equal to four multiplied by three multiplied by two multiplied by one. Okay, great, we know what the notation means. So now, let’s try and work out the problem.
Well, before working on our problem, I just wanna simplify a bit. So let’s think about six factorial. Well, six factorial is six multiplied by five multiplied by four multiplied by three multiplied by two multiplied by one. But we could also write it as six multiplied by five multiplied by four factorial. That’s because four factorial is four multiplied by three multiplied by two multiplied by one. And also this’s gonna be useful in our question because we have four factorial on the denominator. And similarly, we can do the same thing with the second term because we’ve got 27 factorial as the numerator. But the denominator is 28 factorial. And we could write this as 28 multiplied by 27 factorial.
So now, with each term, we can divide through by common factors. So we can see on the left-hand term, we can divide the numerator and denominator by four factorial. And with the right term, then we can divide the numerator and denominator by 27 factorial. And when we do that, we get 30 minus one over 28. And then if we convert 30 into 28s, we’ll have 840 over 28 minus one over 28. Well, we could do this using a calculator by multiplying 30 by 28. But also we can do it using mental method because if you’ve three multiplied by 28, it’s the same as three multiplied by 20 add three multiplied by eight which is equal to 60 add 24 which is equal to 84.
So therefore, 30 multiplied by 28 is gonna be equal to 84 multiplied by 10. And that’s because 30 is 10 times bigger than three. And that gives us the 840 that we were looking for. So therefore, we can say that the expression six factorial over four factorial minus 27 factorial over 28 factorial is equal to 839 over 28. And that’s because 840 over 28 minus one over 28 is 839 over 28. And we’ve left it in the correct way because the question asks us to give our answer as a fraction.