We are asked whether it is possible for r to be greater than n in the formula for the number of combinations, _nC_r. To answer this question, we will first recall the formula.
_nC_r = n!/r!( n- r)!
Let's assume that r is greater than n and take a closer look at the product in the denominator of the formula.
_nC_r = n!/r! (n-r)!
We will now consider the expression (n-r)!. Because n is less than r, the expresion n-r is less than 0.
n < r
⇕
n-r < 0
This means that (n-r)! is the factorial of negative values of n-r. However, factorials are not defined for negative numbers. This follows that r cannot be greater than n in the formula for _nC_r.
Interpretation and Alternative Approach
Recall that _nC_r represents the number of possible ways we can choose r out of n objects at a time without regard to order. Let's again assume that r is greater than n. Although it is not possible for the formula _nC_r, we can still say that there are possible ways we can choose r out of n objects for r greater than n.
r> n ⇒ _nC_r = 0
For example, let's consider r = 5 and n = 3. We want to choose 5 out of 3 fruits to make a smoothie.
There are possible ways we can choose 5 out of 3 fruits.
