# Thread: Sum of 3 digits

1. ## Sum of 3 digits

In how many ways can one select 3 different integers ranging from 1 to 30 inclusive so that their sum is a multiple of 3 (order of selection does not matter)?

(A) 3160
(B) 1360
(C) 1240
(D) 1353
(E) 3240

I don't know where to begin with this one, yeah... I'm pretty much stumped.

2. Please tell us what sorts of tools you have to work with. That is, what topics are surrounding this problem? How are you expected to work it? I ask because I know the correct answer but I had to use the programming ability of a computer algebra system to solve it. I have not found any way to model this problem using counting techniques.

3. Hello, DivideBy0!

I have an approach to this problem, but I don't know if it is valid.
. . If I'm wrong, I hope someone can explain it.

In how many ways can one select 3 different integers ranging from 1 to 30 inclusive
so that their sum is a multiple of 3 (order of selection does not matter)?

. . (A) 3160 . . (B) 1360 . . (C) 1240 . . (D) 1353 . . (E) 3240

There are:
30C3 = 4060 ways to select 3 distinct integers from 1 to 30.

The sums will range from: .1 + 2 + 3 .= .6 .to .28 + 29 + 30 .= .87

Here's where I'm being quite daring . . .
. . I assume that one-third of the sums are divisible by 3.

And 4060 ÷ 3 is suspiciously close to answer (D): 1353.

4. Sorry I didn't reply to your post Plato, the question is from a grade 9-10 competition paper, so the questions are from a wide range of topics.

Originally Posted by Soroban
Hello, DivideBy0!

I have an approach to this problem, but I don't know if it is valid.
. . If I'm wrong, I hope someone can explain it.

There are: 30C3 = 4060 ways to select 3 distinct integers from 1 to 30.

The sums will range from: .1 + 2 + 3 .= .6 .to .28 + 29 + 30 .= .87

Here's where I'm being quite daring . . .
. . I assume that one-third of the sums are divisible by 3.

And 4060 ÷ 3 is suspiciously close to answer (D): 1353.
That's a very elegant way of looking at the problem, and it's very close to the actual answer, which the answers give as (B): 1360. Nice try Soroban.

5. Here the way I did using the Boolean capabilities of a simple CAS.