ⁿCᵣ equals:

The term $^nC_r$ (read as 'n choose r') represents the number of combinations of $n$ distinct objects taken $r$ at a time, where the order of selection does not matter. ### Step-by-Step Derivation: 1. **Permutations ($^nP_r$):** The number of ways to arrange $r$ objects out of $n$ is given by: $$^nP_r = \frac{n!}{(n-r)!}$$ In permutations, the order of the items matters (e.g., AB is different from BA). 2. **Removing Order:** In combinations ($^nC_r$), the order does **not** matter. This means that for every group of $r$ objects we select, there are $r!$ different ways they could have been arranged. To get the number of unique groups, we must divide the total number of permutations by $r!$. 3. **Final Formula:** $$^nC_r = \frac{^nP_r}{r!} = \frac{\frac{n!}{(n-r)!}}{r!} = \frac{n!}{r!(n-r)!}$$ Therefore, the correct expression is $\frac{n!}{r!(n-r)!}$.