# BIGNUM CALCULATOR

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 1st number n: 2nd number r: Calculates to 100 factorial. No fractions. 0 or -1 appears if calculation is not possible.
 Sum: n+r Difference: n−r Product: n*r Quotient: n/r Remainder: n mod r Power: Value range: replaced   ordered nr Factorial: Value range: discarded   ordered n! Permutation: Value range: discarded   ordered P(n,r) Combination: Value range: discarded   unordered C(n,r) Recombination: Value range: replaced   unordered R(n,r)= C(n+r−1,r)

## EXPLANATION

The Bignum Calculator is a Javascript program that uses arrays to calculate numbers beyond the normal reach of hand calculators and computer programs.

The following are explanations of some of the features:

• The Bignum Calculator does only integer math. It can not do decimals or fractions.
• Discarded: This means the objects taken are kept by the taker, or are discarded. They are not put back into the pool to be chosen again.
• Replaced: This means either that the taken objects are put back into the pool, or that there is an infinite number of each kind of object available.
• Ordered: This means the order in which the objects are picked matters. The objects are picked one at a time, and are kept in that order.
• Unordered: This means the objects are chosen as a handful or group. The order in which they are chosen is irrelevant.
• Power: This is multiplying r copies of the number n together.
It tells the number of ways of picking r objects in order from n kinds of objects, with replacement.
Example: Rolling an n-sided die r times.
• Factorial: This is the product of all of the integers from 1 up to the selected number.
It tells the number of ways of arranging n objects in order, with no replacement.
Arranging n different books on a shelf.
• Permutation: This the factorial of n, divided by the factorial of (n-r).
It tells the number of ways of picking r objects from a pool of n different objects, and arranging them in order, without replacement.
Arranging r books on a shelf, selected from a box containing n different books.
• Combination: This the factorial of n, divided by the factorial of (n-r), and then divided by the factorial of r.
It tells the number of ways of choosing a group of r objects, without order, and with no replacement.
Choosing an unranked subcommittee of r people from a group of n.
• Recombination: This the factorial of n+r-1, divided by the factorial of (n-1), and then divided by the factorial of r.
It tells the number of ways to group r objects taken from infinite supplies of n kinds of objects.
The number of possible colors obtainable by mixing r beams of light chosen from n available colors, with replacement. Order does not matter.

Note that the use of recombination in probability is limited. Using the example above:

1. If each available light color is chosen independently, then the available colors have different probabilities. A tree must be used here.
2. If a list of the R(n,r) possible colors is made, and then the random selection is made from that, then R(n,r) can be used in the probability.

## An Explanation of Recombination Probabilities

### What happens if the first color and the second color are chosen independently:

First Color
Second ColorREDGREENBLUE
REDRedYellowMagenta
GREENYellowGreenCyan
BLUEMagentaCyanBlue

If the first color and the second color are chosen independently, the probabilities if the colors are:

• 1/9 Red
• 2/9 Yellow
• 1/9 Green
• 2/9 Cyan
• 1/9 Blue
• 2/9 Magenta

The different colors have different probabilities. All of the table posuition outcomes are equally likely, but the same color can appear in multiple outcomes. Thus, some colors are more likely than others.

### The following method produces equally likely outcomes for all colors:

Use R(n,r) to find the number of possible colors. To find out what the colors are, use the table above, and cross off the duplicate colors.

Make the table below, using the list of colors found, and placing to the right of each color only ONE way to mix that color.

Randomly choose one color from the left column. Then install the colors found in the other columns.

The order the colors are installed in the beams does not matter.

Color ChosenColors Installed
RedREDRED
YellowREDGREEN
GreenGREENGREEN
CyanGREENBLUE
BlueBLUEBLUE
MagentaREDBLUE

The above method produces the following colors in the following probabilities:

• 1/6 Red
• 1/6 Yellow
• 1/6 Green
• 1/6 Cyan
• 1/6 Blue
• 1/6 Magenta

Now all of the colors have the same probabilitiies.