How many possible winning lottery number combinations are there? How many different combinations can be created for a dial lock? These questions and more can be answered with the statistics tools: combinations and permutations. Combinations and permutations are types of calculations used in statistics and precalculus. Learn how to calculate combinations and permutations by hand and with Excel.

Define a combination. A combination is the number of ways that objects can be arranged, in which order of the objects does not matter. This means that AB is not a different way to arrange the items A and B than BA. A good explanation of the order of objects not mattering is ordering a banana split. If the banana split has a scoop of chocolate, vanilla and strawberry ice cream on it, the scoops may be in any order and you still have a banana split with chocolate, vanilla and strawberry ice cream on it.

Objects can be repeated in the arrangement, which is called repetition. In the example above, you might order your banana split with three scoops of chocolate ice cream and no other flavours of ice cream. Objects can also not be repeated in the arrangement. In the example above, you might only have one scoop of each type of ice cream.

Calculate a combination, with repetition, by hand. The variables in the following formula are:

n: total number of objects, referred to as a set.

r: number of objects in arrangement, or combination.

NCR = the number of ways that the objects can be arranged.

!: factorial.

NCR = ( (n + r - 1 ) ! ) / ( r ! ( n - 1 ) ! )

For example:

How many different committees of three students can be formed from a class of five students? Students can be on more than one committee.

n = 5 (the number of objects, or people, referred to as a set).

r = 4 (the number of objects, or people, in each arrangement, or photograph).

NCR = ( (5 + 3 - 1) ! ) / ( 3 ! (5-1) ! ) = 7! / (3! (4!) ) = (7x6x5x4x3x2x1) / ( (3x2x1) (4x3x2x1) )

Calculate a combination, with repetition, using Excel. Use the following formula to calculate combinations, without repetition in Excel, where H1 = n + r -1. F2=r and H2=n-1.

\=FACT(H1)/((FACT(F2)*(FACT(H2))))

Calculate a combination, without repetition, by hand: The variables in the following formula are:

n: total number of objects, referred to as a set.

r: number of objects in arrangement, or combination.

NCR = the number of ways that the objects can be arranged.

!: factorial.

NCR = ( n! ) / (r! (n-r)! )

For example:

How many different teams of four students can be formed from a class of 13 students?

n= 13 (the number of objects, or people, referred to as a set).

r= 4 (the number of objects, or people, in each arrangement or photograph).

n-r+1=13-4+1=10, so you stop the top part of your formula when n-# equals 10.

NCR = 13! / (4! (13-4)! ) = (13 (13-1) (13-2) (13-3)) / (4x3x2x1)

NCR= 715 possible teams.

Calculate a combination, without repetition, using Excel. Use the following formula to calculate combinations, without repetition in Excel, where C1=n and C2=r.

\=COMBIN(C1,C2)

Define a permutation. A permutation is the number of ways that that objects can be arranged, in which the order of the objects matters. This means that AB is a different way to arrange the objects A and B than BA. A good explanation of the order of objects mattering is the example of combination locks. A lock combination 12-31-4-22 is different than a lock combination 4-12-31-22.

Objects can be repeated in the arrangement, or they may not repeat.

Calculate a permutation, with repetition, by hand. The variables in the following formula are:

n: total number of objects, referred to as a set.

k: number of objects in arrangement, or permutation.

NPK: the number of ways that the objects can be arranged.

NPK = n x n x n...(k times)

For example:

On a dial combination lock, there are four dials, numbered 0 through 9. How many combinations are possible?

n= 10 (0-9)

k= 4 (dials)

NPK = 10 x 10 x 10 x 10

NPK = 10,000 possible dial combinations.

Calculate a permutation, with repetition, using Excel. Use the following formula to calculate permutation, without repetition in Excel, where P1=n and P2=k.

\=POWER(P1,P2)

Calculate a permutation, without repetition, by hand. The variables in the following formula are:

n: total number of objects, referred to as a set.

k: number of objects in arrangement, or permutation.

NPK: the number of ways that the objects can be arranged.

!: factorial.

NPK = n! / ( (n-k)! ) = n (n-1) (n-2) ... (n-k+1)

... means that you repeat the pattern of n-1, n-2, n-3, etc. ... until you reach what (n-k+1) is.

For example:

The photographer at your school wants to take pictures of all of the math students but only in groups of five people per picture. There are 18 people in the math class. The example of a photograph is a great example, because one person cannot be in the same photograph more than once.

!

n = 18 (the number of objects, or people, referred to as a set).

k = 5 (the number of objects, or people, in each arrangement or photograph).

n - k + 1 = 18 - 5 + 1 = 14, so you stop your formula when n-# equals 14.

NPK = 18! / 13! = 18 (18-1) (18-2) (18-3) (18-4)

NPK = 1,028,160 possible ways to arrange the students.

Calculate a permutation, without repetition, using Excel: Use the following formula to calculate permutation, without repetition in Excel, where P1=n and P2=k.

\=PERMUT(P1,P2)

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