combination of numbers

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July 22nd 2009, 12:20 AM
tallberg

combination of numbers

Hi

Not sure if im in the correct tread.
Im trying to work out how to create all the collections of numbers that add up to 100.

For example

100*1
or
50+10+40
or
22+22+22+44

Does anyone know how many combinations there are and how to work it out how to make them?
July 22nd 2009, 12:49 AM
bananaxxx

I don't quite understand. You said "add up" to 100. Then you wrote 100*1 (multiplication). Also, if you include negative numbers, there are infinite combinations 120 + (-20) for example...
July 22nd 2009, 01:22 AM
tallberg

Thanks for the reply

To clarify:

Im trying to work out how to create all the collections of positive whole numbers that add up to 100.

For example

1+1+1+1+1 continued to 100
or
50+10+40
or
22+22+22+44

Does anyone know how many combinations there are and how to work it out how to make them?
July 22nd 2009, 07:31 AM
Soroban

Hello, tallberg!

Quote:

Create all the collections of numbers that add up to 100.

For example: . $\begin{array}{c}1 + 1 + 1 + \hdots + 1 \\ 50+10+40 \\ 22+22+22+34 \\ \vdots \end{array}$

Does anyone know how many combinations there are,
and how to work it out, and how to make them?

I will assume that the order of the terms is important.
. . That is: . $(10,90)$ is considered different from $(90,10)$
Otherwise, the problem is extremely complex, solved only in the 1930's.

Consider a marked 100-cm meterstick which we will cut on the marks.

. . $\Box \Box \Box \Box \Box \hdots \Box \Box $

There are 99 marks on the meterstick.
. . For each mark, we have two choices: Cut or No-cut.

Hence, there are: . $2^{99}$ possible choices.

Therefore, there are ${\color{blue}2^{99} \:\approx\:6.3 \times 10^{29}}$ possible collections of numbers.

You can write them out if you like . . . I'll wait in the car.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Let: . $\begin{Bmatrix}0 &=& \text{no cut} \\ 1 &=& \text{cut} \end{Bmatrix}$

We have a list of 99-digit numbers composed of 0's and 1's.

. . $\begin{array}{ccc} \text{Code} & &\text{Addition} \\ \hline 000\hdots00 && 100\\ 000\hdots01 && 99+1 \\ 000\hdots 10 && 98 + 2\\ \vdots && \vdots \\ 111\hdots00 && 1+1+1+\hdots + 3\\ 111\hdots01 && 1+1+\hdots + 2 + 1\\ 111\hdots10 && 1+1+1+\hdots + 2 \\ 111\hdots11 && 1+1+1+\hdots + 1 \end{array}$