10. By permuting the digits of 20130518 , how many different eight-digit positive odd numbers can be formed?
(1 mark)
若把 20130518 的數字重新排列, 可以得到多少個不同的八位正奇數?
(1 分)
10. 3600
10. The two 0 's must not be at the beginning or end. Hence there are $C_{2}^{6}=15$ ways to fix the positions of the 0 's. It remains to permute the remaining six digits, of which two are the same $(1,1,2,3,5,8)$. There are $\frac{6!}{2!}=360$ such permutations. However, since only 4 of the 6 digits are odd, only two-thirds of these will eventually end up with an odd number. Hence the answer is $15 \times 360 \times \frac{2}{3}=3600$.