Fundamental Counting Principles

Fundamental Counting Principle:

In a series of events, the net possible number of ways all events can executed is the product of the possible number of manners each individual event can be executed.

The Bluman text states this multiplication principle 2.

Factorials:

When n is a positive integer, then

n! = n (n-1) (n-2) ... (3) (2) (1)
n! = n (n-1)!
A special case is 0!
0! = 1

Permutations:

The permutation can be defined as an arrangement of objects without repetition where order is very important.

Permutations utilizing all the objects:

The permutation of n objects, arranged to one group of size n, with no repetition, and order being significant is:

nPn = P(n,n) = n!

Illustration: Find out all the permutations of letters ‘ABC’

ABC  ACB  BAC  BCA  CAB  CBA

Permutations of some of objects:

The permutation of n objects, arranged in a group of size r, with no repetition, and order being significant is:

nPr = P(n,r) = n!/(n-r)!

Illustration: Find out all the two-letter permutations of letters ‘ABC’

AB  AC  BA  BC  CA  CB

Shortcut formula for finding a permutation:

Supposing that you begin a n and count down to 1 in your factorials..

P(n,r) = first r factors of n factorial

Distinguishable Permutations:

Many times letters are repeated and all of the permutations are not differentiateable from one other.

Illustration: Find out all the permutations of letters ‘BOB’

To help you differentiate, we will write the second ‘B’ as ‘b’

BOb  BbO  OBb  ObB  bBO  bOB

If we can write ‘B’ as ‘B’, we get:

BOB  BBO  OBB  OBB  BBO  BBO

There are actually only three distinguishable permutations here.

BOB  BBO  OBB

When a word has N letters, k of which are exclusive, and you let n (n1, n2, n3... nk) be the frequency of each of k letters, then the total number of distinguishable permutations is provided by:

N!/(n1! * n2! ... nk!)

Let consider the word "STATISTICS":

Here is the frequency of each letter: S=3, T=3, A=1, I=2, C=1, there are total 10 letters

Permutations = 10!/(3! 3! 1! 2! 1!) = (10*9*8*7*6*5*4*3*2*1)/( 6 * 6 * 1 * 2 * 1) = 50400

Combinations:

The combination can be defined as an arrangement of objects with no repetition where order is not significant.

It must be noted that the difference between a combination and permutation is not whether there is repetition or not -- there should not be repetition with either, and when there is repetition, you cannot use the formulas for permutations or combinations. The merely difference in the definition of permutation and a combination is whether order is much important.

The combination of n objects, arranged in a group of size r, with no repetition, and order being significant is:

nCr = C(n,r) = n!/((n-r)! * r!)

The other way to write a combination of n things, r at a time is utilizing the binomial notation: n over r in parentheses devoid of a fraction bar

Illustration: Find out all two-letter combinations of letters ‘ABC’

AB = BA   AC = CA   BC = CB

The answer is just a three two-letter combinations.

Shortcut formula for determining combination:

Supposing that you begin a n and count down to 1 in your factorials....

C(n,r) = first r factors of n factorial divided by the last r factors of n factorial

Pascal's Triangle:

The combinations are utilized in the binomial expansion theorem from algebra to provide the coefficients of expansion (a+b)^n. They as well form a pattern termed as Pascal's Triangle.

                           1
                         1   1
                       1   2   1
                     1   3   3   1
                   1   4   6   4   1
                 1   5  10   10  5   1
               1   6  15  20  15   6   1
             1   7  21  35  35  21   7   1

Each and every element in the table is the sum of two elements directly above it. Each and every element is as well a combination. The n value is a number of row (start counting at zero) and the r value is an element in the row (start counting at zero). That would make the 20 in next to last row C(6,3) -- it's in the row #6 (that is, 7th row) and position #3 (that is, 4th element).

Symmetry:

Pascal's Triangle describes the symmetric nature of combination. C(n,r) = C(n,n-r)

Illustration: C(10,4) = C(10,6) or C(100,99) = C(100,1)

Shortcut formula for finding a combination:

As combinations are symmetric, when n-r is smaller than r, then switch the combination to its alternative form and then employ the shortcut given above.

C(n,r) = first r factors of n factorial divided by last r factors of n factorial

Tree Diagrams
:

Tree diagrams are a graphical method of listing all possible outcomes. The outcomes are listed in an arranged fashion, therefore listing all the possible outcomes is simpler than just trying to make sure that you have them all listed. It is termed as a tree diagram since of the way it looks.

The first event comes into views on left, and then each sequential event is symbolized as branches off of first event.

The tree diagram to right would illustrate the possible ways of flipping the two coins. The final outcomes are obtained by following each and every branch to its conclusion: They are from top to bottom.

HH  HT  TH  TT

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