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