Question

How many positive integers in the set {50, 51,, . . . , 298, 299} do not contain any even digits?

If you do not mind please explain the steps along the way in detail.

Thank you

Answer 1

There are 299-50+1 = 250 numbers in the set.

Since half the numbers are even, that leaves 125 with odd last digit

There are now 5 numbers, beginning with 5...29, or 25 sets of 5 numbers. There are 13 sets with odd 2nd digit, and 12 sets with even 2nd digit. Throw those 60 away, leaving 65 starting with 5,7,9,...,29

So, now there are 15 2-digit numbers, and 50 3-digit numbers starting with 10,11,...,29

Half of those start with 2,4,...,28, so toss out those 25, leaving 15+25 = 40 numbers with all odd digits.