Math Help!!

[KathiWithAnI]

Ok, my math team has a competition in a couple weeks and while working through some problems today, we came across this problem:

Set s is the set of all positive four-digit integers that have four distinct digits. How many of the integers in the set have a value less than 1235?

Supposedly the answer is 65 integers. I am sure how to solve it is staring me in the face, but I must have tired brain tonight. I'm now wondering if by 'value' I was supposed to add the digits together???
Can anyone sort this one out for me???

Thanks so much!
Kathi

 

[java]

Try to look at it as a shape?1235 as the number of sides with the diagonals being the numbers inside??
Does that make any sense or trigger anything? I can't get it to work though so that might not be even close but that's where my son's mind went.

Oh and by value I read it as any number higher than 1023 but less than 1235? does that make sense?

 

[SAHDad]

Hrm. I'd have to think about it for a bit, but I am assuming (for example) that if you have 1024, you cannot use 1042 (since they have the same 4 digits, albeit rearranged).

With that in mind, I came up with about 30 of them, but I'd have to think about getting to 65.

 

[BuzznBelle'smom]

I don't know what the PP is saying about a shape.

Here's how I understnad the question: You are looking for all 4-digit integers with 4 different digits, that are less than 1235. I don't know the exact answer, but:

1023
1024
1025
1026
1027
1028
1029
1032
1034
1035
...
1234

 

[java]

I don't know what the PP is saying about a shape.

The thinking is that yes you could manually write them all out- but I'm guess there is a formula with diagonals that would work. But I'm not smart enough to know which formula- just assuming there is one. Again though just a guess.

 

[taterules]

This isn't really 1 formula, but you can break it down into parts.

For numbers with a 10xx format, there is:
1 choice for the first digit (it's a 1)
1 choice for the second digit (it's a 0)
8 choices for the third digit (anything but 1 or 0)
7 choices for the fourth digit (anything but 1, 0, or the third digit)
That means 1 * 1 * 8 * 7 = 56 numbers will have the form 10xx.

No numbers will be in 11xx format because you can't repeat the 1.

In the 12xx format, staying under 1235, break it down to:
120x has 7 choices for the fourth digit
121x doesn't work (repeats the 1)
122x doesn't work (repeats the 2)
123x has 2 choices (just 0 or 4 will work and be under 1235).

So, 56 choices for 10xx, 7 choices for 120x, and 2 choices for 123x. That's 65 total.

I hope I explained that clearly.