13Feb/091

## MMM #25: Winner!

Ron Frederick is the winner of MMM #25: Monotonous Monday Math Madness. Congratulations, Ron!

Jonathan had a very simple explanation, based on elementary combinatorics:

Line up the digits 123456789 and take any one of them: C(9,1) choice.

Take any two of them: C(9,2).

Now, wait. Am I saying there are C(9,2) = 36 2-digit ascending numbers? 12, 13, 14, 15, 16,17, 18, 19, 23, 24, 25, 26, 27, 28, 29, 34, 35, 36, 37, 38, 39, 45, 46, 47, 48, 49, 56, 57, 58, 59, 67, 68, 69, 78, 79, 89. Yup, thirty-six of them.

So we have [ C(9,1) + C(9,2) + C(9,3) + C(9,4) + C(9,5) + C(9,6) ] / 1,000,000 = [9 + 36 + 84 + 126 + 126 + 84]/1,000,000 = .000465

Blinkdagger has the next MMM so check their site on Monday.

SaraFebruary 19th, 2009 - 16:41

While this solution is correct, I think it might have been a little better if it had explained WHY it is true.

For those interested, if you are trying to figure out how many 2-digit ascending numbers, Ron tells you to calculate C(9, 2) = 36. 36 is the number of two-digit subsets you can form from the numbers 1 – 9. So why is the number of subsets also the number of 2-digit ascending numbers? Each of these subsets is unique and can only be ordered with the digits increasing one way. So, take each 2-digit subset and order it correctly!

Hope that helps!