How many integers between $1000$ and $9999$ inclusive consist of
(a) Distinct odd digits,
(b) Distinct digits,
(c) From the number of integers obtained in (b), how many are odd integers?
(a) ${}^5P_4 = 120$
(b) $9 \times 9 \times 8 \times 7 = 4536$
(c) I don't know how to find the odd integers in $4536$
Are the answers correct? Can you guys help solve me the (c)?
$\endgroup$1 Answer
$\begingroup$If you place $0,2,4,6,$ or $8$ in the units place, it becomes even. Hence, you will have to place $1,3,5,7,$ or $9$ in the units place to make it odd. So in the units place, you have $5$ choices. In the thousand's place we have $8$ choices ($8$ because we have left out $0$ and the number in the units place). Similarly we have $8$ and $7$ choices in the hundreds and tens place, respectively . Hence, the answer should be $8\times 8 \times 7 \times 5 = 2240$.
$\endgroup$ 5
