Simplify the expression 28 factorial over 27 factorial minus 46 factorial over 47 factorial, giving your answer as a fraction.
Well, the first thing we need to understand is what factorial means. Well, factorial is this thing that looks like an exclamation mark. And if we have 𝑛 factorial, this is equal to 𝑛 multiplied by 𝑛 minus one multiplied by 𝑛 minus two, etcetera, all the way down to when we get to multiplied by one. For example, three factorial is equal to three multiplied by two multiplied by one. So if we consider our expression, what we need to do first is rewrite it. And we’re gonna rewrite it by splitting our factorials.
So first of all, we can start with first term. And our numerator can be 28 multiplied by 27 factorial. And that’s because 28 multiplied by 27 multiplied by 26, etcetera, will be 28 factorial. So, therefore, it’s the same as 28 multiplied by 27 factorial. But why have we done this? Well, the reason we’ve done this is because the denominator is 27 factorial. So, therefore, they’re going to be able to cancel. So 28 factorial over 27 factorial is just gonna be left as 28. And that’s because the factorials of 27 on numerator and denominator will cancel. And then we can use a similar method for the right-hand term. And that’s because we’ve got 46 factorial over 47 multiplied by 46 factorial.
So this time, the 46 factorials can cancel out. So we’re gonna be left with 28 minus one over 47. So now, what we need to do is just simplify this because we want the answer left as a fraction. Well to work this calculation out, what we need to do is we need to have both of our fractions with the same denominator. So what we need to do is turn 28 into 47. Well to do that, we’ll multiply 28 by 47 which gives us 1316. So then we have 1316 over 47 minus one over 47 which brings us to our final answer. And that is that if we simplify the expression 28 factorial over 27 factorial minus 46 factorial over 47 factorial, then the answer is 1315 over 47. And that’s left as a fraction.