5 digit odd numbers can be made using the digits 0,1,2,3,4,5,6,7 and 8 with first digit being non zero and the digits can't be repeated

Question

I was questioned, that there are 5 digit odd numbers can be made using 0 to 8 and the first digit cannot be zero and the repetition are not allowed of the digit. Also they all have to be odd numbers.

Answer 1

Well, your restrictions are that the last digit must be odd, and the first digit cant be zero. So choose those first. The most restrictive is the last digit must be odd so choose that first, then choose the first digit, and then choose the remain three digits in any order you want.

So there are $4$ choices for the last digit. After choosing that one digit there will be $8$ remaining digits to choose for the first digit but the first digit can't be $0$ so there are $7$ choices.

After those two choices there are $7$ choices for the second digit. After those three choices there $6$ choices for the third digit. And after those four choices there $5$ choicer remaining for the fourth digit.

So there are $4*7*7*6*5$ possible outcomes.

Answer 2

Odd number: units digit must be $1, 3, 5, 7$.

First number non-zero: $7$ choices left (after choosing the units digit)

Next three numbers have $7*6*5$ choices.

Total: $4 \cdot 7 \cdot 7 \cdot 6 \cdot 5 = 5880$