How likely are we going to get "descending numbers"?

Number Theory Level 2

Use numbers $0, 1, 2, 3, 4, 5$ to create non-repeating four-digit numbers. If we call the number with every digit smaller than the one on its left " descending number ", what is the probability for the four-digit number to be a "descending number" ?

\frac {1}{25}

\frac {1}{4}

\frac {1}{14}

\frac {1}{20}

1 solution

Kevin Xu
Sep 9, 2019

First we put six number in order like this: $543210$ $\\$ We can see that it is a six-digit "descending number". $\\$ All we need to do now is to take out two digits to make the number four-digit $\\$ Descending numbers: $C^2_6 = 15$ $\\$ All arrangements $A^1_5 \cdot A^3_5 = 300$ [Number cannot starts with zero, so there are five choice for the first number, then pick 3 out of the rest 5] $\\$ $\frac {15}{300} = \frac 1{20}$

Hi, I think there has been a mistake. All the arrangements should be ${6} \choose {4}$ $\times 4!$ . Also, you may think about the problem like this: with equal probability, we can choose $4$ out of the $6$ numbers and, regardless of the numbers, there would be only one descending order for the chosen $4$ . so, the answer $\frac{1}{4!}$

A Former Brilliant Member - 1 year, 9 months ago

That's a neat thought, it is just that you forgot that a number cannot start with 0 in the first place. Taking that into account would get us 1/20. (Btw, my previous solution does have some problems, it has been updated.

Kevin Xu - 1 year, 9 months ago

Oh, I did not take that into account

A Former Brilliant Member - 1 year, 9 months ago

I got the same answer as kevin, 300 numbers, 15 of which are descending
my only complaint is the fractions arent showing up properly in the answer, maybe a staffer can fix those

Kyle T - 1 year, 9 months ago

Could you please include the code, you have written, as a solution? Your codes provide insights into the problems

A Former Brilliant Member - 1 year, 9 months ago

https://kyletripp.com/code-samples/how-likely-are-we-going-to-get-descending-numbers/

Kyle T - 1 year, 9 months ago

How likely are we going to get "descending numbers"?

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