If seven choose 𝑟 is greater than one and 𝑟 choose five is greater than one, find six minus 𝑟 factorial.
To answer this question, it’s sensible to begin by evaluating seven choose 𝑟 and 𝑟 choose five. By doing this and comparing these expressions to the given inequalities, we should be able to find the value of 𝑟 which we can then use, in turn, to find the value of six minus 𝑟 factorial. At this point, it might be worth noting that some regions use a slightly different notation for factorial. Some viewers might be more familiar with six minus 𝑟 factorial as shown.
The combination formula tells us the number of ways of choosing a sample of 𝑟 elements from 𝑛 distinct objects. And it’s usually denoted as 𝑛 choose 𝑟 as shown. It’s given as 𝑛 factorial over 𝑟 factorial multiplied by 𝑛 minus 𝑟 factorial. This means that we can evaluate seven choose 𝑟 as seven factorial over 𝑟 factorial multiplied by seven minus 𝑟 factorial. We are told that this must be greater than one. For this to be the case, 𝑟 must be less than seven. This will mean we are guaranteed that the numerator will be larger than the denominator of our fraction. We can therefore say that seven choose 𝑟 will be greater than one.
If you’re struggling to see this, try a few examples. We could let 𝑟 be equal to one. Seven choose one could be calculated by finding the factorial of seven and then dividing that by one factorial multiplied by six factorial. By definition, that must be greater than one. If 𝑟 was equal to two, seven choose two would be seven factorial over two factorial multiplied by five factorial, still greater than one.
Let’s repeat this process for 𝑟 choose five. 𝑟 choose five is given as 𝑟 factorial over five factorial multiplied by 𝑟 minus five factorial. And we’re told that this must also be greater than one. This time, 𝑟 must be greater than five to ensure that the numerator is, once again, larger than the denominator of the fraction. We have shown that 𝑟 must be greater than five but also less than seven.
Remember, the factorial function is only defined over the natural numbers. Those are positive integers. So we can say that 𝑟 must be equal to six. This means that six minus 𝑟 factorial is actually six minus six factorial. Six minus six is zero. So we’re evaluating zero factorial which is equal to one. So in this question, six minus 𝑟 factorial is one.