Identify the objects that you are trying to figure out the possible arrangements or combinations.
The formula for NCR uses the following variables:
N= the total number of objects, known as a set.
R= the number of objects in a combination, or an arrangement.
NCR = the number of ways that the objects can be arranged.
! = Factorial, which is defined as the product of all positive integers from 1 to a given number.
Use the following written form of the formula to begin calculations:
NCR = ( (n + r - 1 ) ! ) / ( r ! ( n - 1 ) ! )
Use the following example by which to understand how to apply the formula to your situation.
Example:
How many different groups of three individuals can be formed from a group of five group members? Individuals can be in more than one group.
Extract the information for each variable provided in the example.
n = 5 (the number of objects, or people, known as a set).
r = 4 (the number of objects, or people, in each arrangement, or combination).
NCR = ( (5 + 3 - 1) ! ) / ( 3 ! (5-1) ! )
Now calculate each parenthesis from right to left. Since you can't list the factorials yet you will need to add a symbol after each calculation is complete
( (5 + 3 - 1) ! ) / ( 3 ! (5-1) ! ) = 7! / (3! (4!) )
Continue to calculate until you no longer have a factorial. The factorial is all positive numbers. Begin with the first number and list all positive numbers in descending order.
7! / (3! (4!) ) = (7x6x5x4x3x2x1) / ( (3x2x1) (4x3x2x1) )
Produce the final number of possible combinations of grouping the individuals. Parenthesis means multiply the slash (/) means divide.
(7x6x5x4x3x2x1) / ( (3x2x1) (4x3x2x1) )
5040 / (6x24)
5040/144=35
NCR=35 possible combinations of groupings.
Identify the objects that you are trying to figure out the possible arrangements or combinations.
The variables in the formula are the same however the actual written form is as follows.
nCr = ( n! ) / (r! (n-r)! )
Use the following example by which to understand how to apply the formula to your situation.
Example:
How many different groups of four individuals can be formed from a group of thirteen group members? Each individual can only be in one group.
Extract the information for each variable provided in the example.
n=13 (the number of objects, or people, known as a set).
r= 4 (the number of objects, or people, in each arrangement or combination).
Because there can be no repetition you must isolate that in the equation in this manner.
n-r+1=13-4+1=10, so you stop the top part of your formula when n-# equals 10.
Calculate each parenthesis from right to left. Since you don't can't list the factorials yet you will need to add symbol after each calculation is complete. Notice that the calculation for '4! was moved to the end of the equation for ease of calculation and the 13-4! was broken down in descending or from 4 these are the factorials in this equation.
13! / (4! (13-4)! ) = (13 (13-1) (13-2) (13-3)) / (4x3x2x1)
Produce the final number of possible combinations of grouping the individuals. Parentheses mean multiply; the slash (/) means divide.
13 x (12x11x10=1320) / (4x3x2x1)=24
(13x1320)/24
17160/24=715
NCR= 715 possible groups.