Given $5$ distinct numbers $1,2,3,4,5$. without repetitions, find number of $4$ digit odd numbers.
For odd numbers, the last number must be odd $(1,3,5)$ and one number must be picked for the last number. Thus ${}_3C_1$.
The remaining $3$ front numbers, any other number can be picked except for one odd number. So with that, we have only $4$ options to choose from and we must pick $3$.
However, my answer booklet put it as ${}_4P_3$.
Therefore, ${}_4P_3 \times {}_3C_1 = \text{answer}$.
Why is it "$P$"? Permutations ($P$) means that the order matters. why does the order matters? I thought as long as the last digit is odd means the number is odd?
The order matters because the numbers $2345$ and $4235$ are different numbers.