4.3 Permutations with Some Identical ElementsThis is a featured page


There are three different type of permutations:

a) Permutations of all elements; P(n,n) = n!

Example: What are the total number of arrangements for the following six balls if all
balls must be used?

4.3 Permutations with Some Identical Elements - MDM4U1@FMG

b) Permutations involving some elements from the population;4.3 Permutations with Some Identical Elements - MDM4U1@FMG

Example: What are the total number of arrangements for the following six balls if only
three balls must be used?

4.3 Permutations with Some Identical Elements - MDM4U1@FMG

These two types were demonstrated in the previous lesson.

c) Permutations involving some identical elements; 4.3 Permutations with Some Identical Elements - MDM4U1@FMG

Example: What are the total number of arrangements for the following: 3 blue balls, 2 red balls, and 1 green ball?

4.3 Permutations with Some Identical Elements - MDM4U1@FMG

Explanation:

The
total number of permutations for the arrangement of the six balls is 6!, but these are not distinct permutations. The distinction between the permutations within three blue balls and also between the two red balls is not possible, resulting in repetition which includes non-distinct permutations. In order to eliminate this repetition, the total number of permutations are divided by the permutations of the repeated elements.

Therefore, resulting in:

4.3 Permutations with Some Identical Elements - MDM4U1@FMG

Example 1: How many distinct permutations can be made from the name AJAY?

The total number of permutations for the name AJAY are 4!.

These are not distinct permutations, the permutations of the identical letters (elements), “A”, do not affect the total number of distinct permutations possible.

Therefore, the number of distinct permutations for AJAY does not equal 4!

To eliminate the non-distinct permutations, the total number of permutations are divided by the permutations of the repeated identical elements.

Therefore, the total number of distinct permutations for AJAY are:
4.3 Permutations with Some Identical Elements - MDM4U1@FMG

Example 2:
How many 7-digit even numbers less than 3,000,000 can be formed using the following digits: 1, 2, 2, 3, 5, 5, 6?

This questions involves cases:

Case 1: First digit is 1 and last digit is 6.

4.3 Permutations with Some Identical Elements - MDM4U1@FMG

Case 2: First digit is 2 and last digit is 6.

4.3 Permutations with Some Identical Elements - MDM4U1@FMG

Case 3: First digit is 2 and last digit is 2.

4.3 Permutations with Some Identical Elements - MDM4U1@FMG

Case 4: First digit is 1 and last digit is 2.

4.3 Permutations with Some Identical Elements - MDM4U1@FMG

Now use the Additive Counting Principle (ACP): (4.1 Organized Counting)

4.3 Permutations with Some Identical Elements - MDM4U1@FMG
Therefore, there are a total 210 distinct permutations.

The process for finding the number of permutations can be summarized in this flow chart:

4.3 Permutations with Some Identical Elements - MDM4U1@FMG


This is a video of
some of the permutations for a Rubik cube that lead to the solution of the cube.


References:

Mathematics of Data Management , Grade 12, (MDM4U). McGraw-Hill Ryerson
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Anonymous How would this be tackled using your line of reasoning 2 Nov 22 2012, 1:03 PM EST by Anonymous
 
Thread started: Aug 21 2012, 3:16 PM EDT  Watch
A group of 5 friends—Archie, Betty, Jerry, Moose, and Veronica—arrived at the movie theater to see a movie. Because they arrived late, their only seating option consists of 3 middle seats in the front row, an aisle seat in the front row, and an adjoining seat in the third row. If Archie, Jerry, or Moose must sit in the aisle seat while Betty and Veronica refuse to sit next to each other, how many possible seating arrangements are there?
32
36
48
72
120
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Anonymous math 0 Aug 7 2012, 10:40 PM EDT by Anonymous
 
Thread started: Aug 7 2012, 10:40 PM EDT  Watch
9!/4!5!
please ans me
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Anonymous xfcgf 0 Apr 10 2012, 2:34 PM EDT by Anonymous
 
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yhgdrfh
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