# General formula for a combination of four categories

• AirRecce
In summary, the problem is to find a general expression for the possible number of unique "deals" that can be formed from a given set of "I" properties, "J" investors, "K" mortgages, and "L" credit lines, where "I" and "J" must be integers greater than zero and "K" and "L" are non-negative integers. The solution may involve multiplying the number of options for each category, rather than using a summation approach.
AirRecce

## Homework Statement

Say I have four categories which make up a "whole" that I'll call a unique "deal".

Each deal can have "I" properties, "J" investors, "K" mortgages, and "L" credit lines, where "I" and "J" must be integers greater than zero and "K" and "L" are non-negative integers (i.e. 0 or positive integers). How do I write a general expression to find out the possible number of unique "deals"?

## Homework Equations

For a combination with a set number of items selected from a set whole, we would use the formula:

(n!) / (r!(n-r)!) , where r = number of objects pulled from a set of n objects

But in this case, we want any combination of the above, not just a set number of r objects. So I think the answer depends on a summation of all the possible values of r from 1-n for the set of n objects

## The Attempt at a Solution

set = I + J + K + L = n
objects = r

so Σ = [ ((n!) / (1!(n-1)!)) + ((n!) / (2!(n-2)!)) ... ((n!)/(n!(n-n)!) ]
r=1

But this is including cases where there are no investors or properties in some of the combinations, which I don't want.

If I understand you correctly I think you just multiply the number of each valid option. For example if "I" can take one of 9 values, "J" can take one of 5 values and ,"L" one of 4 values the number of combinations is

9 * 5 * 4 ...

PS You have a problem if some values can be any positive integer because there are an infinite number of positive integers.

AirRecce said:

## Homework Statement

Say I have four categories which make up a "whole" that I'll call a unique "deal".

Each deal can have "I" properties, "J" investors, "K" mortgages, and "L" credit lines, where "I" and "J" must be integers greater than zero and "K" and "L" are non-negative integers (i.e. 0 or positive integers). How do I write a general expression to find out the possible number of unique "deals"?
I think part of your confusion stems from not having a complete statement of the problem. Is J the total number of possible investors, so that the number of investors in a given deal is j∈{1, 2, . . . J}?
Is a sum the right way to determine combinations? If you have for example, 2 possible combinations of properties, 3 possible combinations of investors, 1 possible combination of credit lines, 1 possible combination of mortgages, how many possible combinations of properties, investors, credit lines and mortgages would you have?

## 1. What is a general formula for a combination of four categories?

The general formula for a combination of four categories is nCr = n! / r!(n-r)!, where n represents the total number of items in the set and r represents the number of items chosen for the combination.

## 2. How is this formula derived?

This formula is derived from the principles of combinatorics, specifically the concept of combinations. It is based on the number of ways to choose r items from a set of n items, without regard for order.

## 3. Can this formula be used for any type of problem involving four categories?

Yes, this formula can be used for any type of problem involving four categories, as long as it follows the principles of combinations. It can be applied to a variety of fields, such as mathematics, statistics, and computer science.

## 4. Are there any limitations to using this formula?

There are some limitations to using this formula, as it assumes that all items in the set are distinct and that each item can only be chosen once for the combination. It also does not take into account any restrictions or conditions in the problem.

## 5. How can this formula be applied in real-life situations?

This formula can be applied in real-life situations such as calculating the number of possible combinations of ingredients in a recipe, the number of possible outcomes in a game, or the number of possible ways to arrange a group of people in a seating chart.

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