Question

1. You have the digits 0-9. How many 6-digit permutations are odd and do not start with 0

Answer 1

We have 10 digits to choose from (0-9), so the total number of 6-digit permutations is 10 × 10 × 10 × 10 × 10 × 10 = 1,000,000.

Of these, the permutations that start with 0 can be eliminated, leaving 9 × 9 × 9 × 9 × 9 × 9 = 729,000.

Now, we need to figure out how many of these are odd. To do this, we note that odd numbers end in 1, 3, 5, 7, or 9. This means that the last digit of an odd 6-digit permutation is always one of these five digits. Since we are choosing the last digit from 5 options, there are 5 × 729,000 = 3,645,000 odd 6-digit permutations that do not start with 0.