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Permutations and Combinations











































Problem: In how many ways can a set of 12 unlike objects be divided into 2 sets of 6?

Comments for Permutations and Combinations

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May 17, 2011
Permutations and Combinations
by: Staff


The question:

Problem: In how many ways can a set of 12 unlike objects be divided into 2 sets of 6?


The answer:


Your question title is “Permutations and Combinations”.


However, your problem does state how the order of the unlike objects should be taken into account. Does the order of the objects make a difference?

Are the arrangements {1,2,3,4,5,6} and {7,8,9,10,11,12} considered the same as {6,5,4,3,2,1} and {12,11,10,9,8,7}?


If the order (the sequential arrangement of the individual objects) does not matter, it called a Combination.

If the order (the sequential arrangement of the individual objects) is important, it called a Permutation. The permutation is a list of all possible ways of choosing. For example, if you have a list of 3 letters, the permutation will tell you how many different ways you can arrange the letters. The permutations are = 3! = 3*2*1 = 6 combinations:

a, b, c
b, c, a
c, a, b
a, c, b
b, a, c
c, b, a

Of course all six possibilities include the same letters (a, b, and c) arranged in a different order.

If order does not matter, there is only 1 way of making the selection: 3!/3! = 3*2*1/3*2*1 = 1

If order does not matter, then a, b, and c is the same as b, c, and a; etc.


The number of permutations for your group of 12 objects is:

12! = 12*11*10*9*8*7*6*5*4*3*2*1 = 479,001,600

The number of combinations groups of 6 can be chosen from the 12 objects is:

12_C_6 = 12!/6!

= 12*11*10*9*8*7

= 665,280

IF two groups of 6 are chosen

= 12!/(6!6!)

= 12*11*10*9*8*7/(6*5*4*3*2*1)

= 924







Thanks for writing.

Staff
www.solving-math-problems.com




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