Gifted Issues Discussion homepage Forums General Discussion "Numbers to 1000 whose digits equal 4?"

My daughter has a math problem asking her to list all the whole numbers from 0 to 1000 whose digits equal 4.

I don't know how to do this problem, and unfortunately, I'm not seeing explicit instructions on how to approach this problem.

Can anyone suggest how to approach solving this problem? (Resources also appreciated.)
Thank you!

Is the problem asking for the numbers whose digits *add* up to 4? Or something else?

polarbear

By "digits equal 4," do you mean that the digits sum to four?

What you want in that case is to first figure out all possible combinations of digits between 0 and 4 that add up to four: 0004, 0013, 0022, etc. Then for each set, you want the number of ways to order the digits. Add it all up and you have the answer.

Fun problem!

Thank you so much, Elizabeth! (The way you stated the problem was what I was trying to communicate.)

Oops, had to stop and try it out. Now I got an answer, and it makes me think I missed a slicker solution.

Find the coefficient of x^4 in the MacLaurin series of 1/(1-x)^3.

Why? What's the connection?

An equivalent problem is to find the number of ways to put 4 identical balls in 4 different containers. You can probably find this in any book on intro combinatorics. Then the answer is 7 choose 4, which is 35.

Here is an example:
http://mathforum.org/library/drmath/view/56226.html

AFAIK, it is 'number of ways to place 4 identical balls into 3 containers' = 'throw 2 sticks between 4 balls, on the outside allowed' = C(6,2) = (6*5)/(2*1) = 15.

(3 containers, and not 4, because the number < 1000 and so has <= 3 digits.)

I'm going to guess that this is a mathematician joke; it works (you can see this if you look at the mathforum link someone gave above, plus the definition of a Maclaurin series), but the deep reason why is that 1 - 0 = 1. If 22B can tell us differently, though, e.g. give an interpretation for x that makes an interesting generalisation, I'm all ears!

Ah of course. For some reason I misread it as a 4 digit number.

We don't know the age and background of the OP's daughter, but it looks like the question is designed for someone with no particular background in combinatorics (counting techniques). Instead, the approach is to just try and list all the numbers systematically, e.g. in increasing order. (The problem said to actually list them, not just count them.) It's most educational if the student just tries to figure it out by themselves without being taught a method.

In other words there's not supposed to be an already-learned method, and I was trying to make this point in an obscure way by giving a (counting) method.

If you want a counting method, a classic one is given in the mathforum.org link above. Another classic counting method is the "generating function" method I gave above. Briefly, you want to count solutions to
a+b+c=4 (a,b,c non-negative integers)
Now 1/(1-x)=1+x+x^2+x^3+x^4+x^5+...
Distribute
(1+x+x^2+x^3+x^4+x^5+...)(1+x+x^2+x^3+x^4+x^5+...)(1+x+x^2+x^3+x^4+x^5+...)
and collect like terms.
The fifteen x^4 terms correspond to all the products (x^a)(x^b)(x^c)=x^{a+b+c} equalling x^4.

A lovely, succinct explanation. Thank you!