Question

In how many ways can you write 18 as the sum of three counting numbers?

Answer 1

hmm what class is this for?

Answer 2

algebra 2/Trig

Answer 3

Is this a homework problem, or are you just curious? This is a question for combinatorics.

Answer 4

this is review for the test tomorrow. I think my teacher is trying to start us on pre calc for next year so i think thats what this is.

Answer 5

Do you have any idea how you are supposed to solve this?

Answer 6

on these type of combination problems we've been using factorials, and sometimes exponential functions, but I don't know where to start on this one

Answer 7

3! = 6
6 + 6 + 6
10+7+1
10+5+3
10+6+2
.......

Answer 8

are counting numbers just natural?

Answer 9

so its 3 ! ?

Answer 10

I can list much more than three if I understand the question right. But I don't think I do

Answer 11

err much more than 6

Answer 12

we've also been using pascals triangle if that helps.

Answer 13

@Luis_Rivera can you explain please

Answer 14

is it possible that it has something to do with the 18th row? or 18!/(3!15!) or something?

Answer 15

I have no idea, maybe luis will answer.

Answer 16

haha I hope so

Answer 17

but I can list way more than 6 ways of writing 18 with the sum of three natural numbers

Answer 18

10 + 5 + 3= 5 *2 + 5 + 3
primez : 5, 3, and 2

6 + 4 + 8 = 3*2 + 2*2 + 2^3
And on and on,

notice that even though you see a 5, the actual numbers used total 3.
Anyway that's how I was taught in ancient India

Answer 19

ahh I see.

Answer 20

so its how many numbers are used, not 3 numbers added together.

Answer 21

bah im still confused...lol gl @liv2dance

Answer 22

We seem to be ignoring repeats. it does NOT say 3 DISTINCT values.

Try expanding this \(\left(\sum\limits_{n = 1}^{16}x^{n}\right)^{3}\), and just reading the coefficient off the resulting \(x^{18}\) term.

I get 136.

Answer 23

I know it does not say distinct values, but they must be assumed, like in permutations of a word. Even though they do not say distinct you will not use the same word twice

Answer 24

i think @tkhunny is right though.

Answer 25

I don't see what the prime numbers have to do with finding the amount of (__+__+__) that equals 18

Answer 26

Generating function can be tedious, to be sure, but the idea is not overwhelming.

Answer 27

Haha sorry. I'm almost positive @tkhunny is right. Thanks for everyones help though