What is the probability that a randomly chosen integer between 1 & 1,000,000 inclusive has the property:?

Question

that no digit is smaller than a digit to its left?
Prove your answer. Computer generated solutions will not be regarded.

Answer 1

Here's maybe a simpler way of looking at the problem (than aa_b's correct solution):

First, how many such numbers are there between 000000 and 999999 inclusive? It's just:

(9+6)!/(9!6!) = 5005

One way to see this is to represent each qualifying number pictorially by a sequence of dots (6 in total) and bars (9 in total). There's a bijection between the set of all qualifying numbers, and the set of all such permutations of 15=6+9 symbols, since the bars divide the dots into 10 compartments, with the number of dots in each compartment giving the number of 0's, 1's,...,9's respectively. For example:

......||||||||| --> 000000
|||||||||...... --> 999999
etc.

To use this result for the actual set, 000000 (which satisfies the criterion) is replaced by 1000000 (which doesn't), so the desired total is 5005-1 = 5004, giving the probability:

5004/1000000 = 1251/250000

Answer 2

let me try, But this may be the most ugly solution:

1-10 all nine single digital integer are ok. (1+1+1...=9)

11-100 there are: (9+8+7+...1 = 9*(9+1)/2=45)
11,12...19 (9)
22,23,...29 (8)
33,34,...39 (7)
...
99 (1)

101-1000 there are 45+36+28+21+15+10+6+3+1=165

111, 112, 113,...119 (9)
122, 123, 124,...129 (8)
...
199 (1)

222, 223,224,...229 (8)
233,...........................(7)
244............................(6)
..
299 (1)
----------------------------------------------
1001-10000
(165+120+84+56+35+20+10+4+1)=495

1111-1999 (165)
2222-2999 (165-45)=120
3333-3999 (165-45-36)=84
......
9999
-------------------------------------------------------------
10,001-100,000
495+330+210+126+70+35+15+5+1
=1287

100,001-1,000,000
1287+792+462+252+126+56+21+6+1
=3003

Total number=9+45+165+495+1287+3003
=5004

Answer 3

I will outline the general idea

Consider the number is abcde

Now a cannot be 0

b >=a
c>=b
d>=c
e>=d

a can take up any of the values between 1 to 9

And the rest shall follow.

Answer 4

is this an integer?

hehe, should have read the question a little more closely :)

I'm not really seeing an easy way to show this. Maybe I'll get back to you later.

Answer 5

Man... now I wish I'd paid more attention to math in school!

Answer 6

Is that my left, or "stage" left?