In order to make sure that the number will be odd, you must choose an odd digit for . There are 5 odd digits, but since the digit 5 must not occur anywhere in the number you are about to construct you have only 5-1=4 possibilities for .
For there is no restriction placed on the digit, except that it must not be 5. So there are 10-1=9 possibilities for your choice of .
Now for digit : again, it must not be 5 but also, since it is the leading digit of your number, you must not make it 0 either. So there are 10-1-1=8 possible choices for .
Since these choices are independent of each other, the total number of possibilities is the product
Again, imagine yourself selecting the 3 highest placed persons by yourself. For the first place, you have 8 possibilities. For each of these 8 possibilities you have 8-1=7 possibilities to choose a person for the second place. And, finally, you have 8-2=6 possibilities to choose a person for the third place. This gives different ways in all.2. A talent contest has 8 contestants. Judges must award prizes for first, second, and third places. If there are no ties, in how many different ways can the 3 prizes be awarded?