Theory of Counting Principles

Counting Principles:

All branch of mathematics consists of its own fundamental theorem(s). When you check out the fundamental dictionary, we will see that it relates to the foundation or base or is elementary. Fundamental theorems are significant foundations for the rest of material to follow.

Given below are some of the basic theorems or principles.

Fundamental Theorem of Arithmetic:

Each and every integer more than one is either prime or can be stated as an unique product of prime numbers.

Fundamental Theorem of Algebra:

All polynomial in one variable of degree n > 0 has at least one real or complex zero.

Fundamental Theorem of Linear Programming:

When there is a solution to linear programming problem, then it will take place at a corner point, or on the line segment among two corner points.

Fundamental Counting Principle:

When there are m ways to do one thing, and n ways to do the other, then there are m*n ways of doing the both.

The Fundamental Counting Principle is the guiding rule for determining the number of ways to achieve two tasks.

Illustrations using the counting principle:

Let's state that you want to toss a coin and roll a die. There are two ways that you can toss a coin and 6 ways that you can roll a die. There are then 2 x 6 = 12 ways that you can toss a coin and roll a die.

When you want to press one note on a piano and play one string on a banjo, then there are 88 * 5 = 440 ways to do the both.

When you wish to draw 2 cards from a standard deck of 52 cards devoid of replacing them, then there are 52 ways to draw first and 51 ways to draw the second, therefore there are total of 52*51 = 2652 ways to draw two cards.

Sample Spaces:

The listing of all possible outcomes is termed as the sample space and is represented by the capital letter S.

The sample space for the experiments of tossing a coin and rolling a die are S = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}.

Sure adequate, there are twelve possible ways. The fundamental counting principle permits us to figure out that there are twelve ways devoid of having to list them all out.

Permutations:

The permutation is an arrangement of objects, devoid of repetition, and order being significant. Other definition of permutation is a number of such arrangements which are possible.

18_permutations.jpg

As permutation is the number of ways we can arrange objects, this will always be a whole number. The denominator in the formula will always divide uniformly to the numerator.

The n value is the total number of objects to choose from. The r is a number of objects you are actually employing.

The two key things to note regarding permutations are that there is no repetition of objects permitted and that order is significant.

Illustrations of permutations:

Illustration: List all the permutations of the letters ABCD

1910_permutation_1.jpg

Now, if we didn't really need a listing of all the permutations, we could use the formula for the number of permutations. There are 4 objects and you are taking 4 at a time. 4P4 = 4!/(4-4)! = 4!/0! = 24/1 = 24.

This as well gives us the other definition of permutations. The permutation whenever you comprise all n objects is n!. That is, P(n, n) = n!

Illustration: List all the three letter permutations of letters in the word HAND

1709_permutation_2.jpg

Now, when you did not actually require a listing of all permutations, you could utilize the formula for the number of permutations. There are 4 objects and you are taking 3 at a time. 4P3 = 4! / (4-3)! = 4!/1! = 24/1 = 24.
Finding Permutations by Hand:

By hand, we can plug the values for n and r to the expression comprising factorials and then simplify the ratio of factorials.

Though, there will always be n-r terms in common between the numerator and denominator once the factorials are expanded. Such terms will divide out, leaving you with first r terms of expansion in the numerator. This provides us a shortcut for determining a permutation by hand.

nPr = first r factors of n!

Finding Permutations with the Calculator:

There is a permutation function on calculator. On TI-82 and TI-83, it is found beneath the Math menu, the Probability Submenu and then choice 2. This is shown as nPr. Enter the value for n first, and then the function, and lastly the value for r.

Combinations:

The combination is an arrangement of objects, with no repetition, and order not being significant. The other definition of combination is the number of such arrangements which are possible.

2079_combinations_1.jpg

The n and r in the formula signifies for the total number of objects to select from and the number of objects in the arrangement, correspondingly.

The key points to a combination are that there is no repetition of objects permitted and the order is not significant.

List all the combinations of letters ABCD in groups of 3.

There are merely four combinations (ABC, ABD, ACD and BCD). Listed below each of such combinations which are six permutations which are equivalent as combinations.

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We know that the combinations were symmetric. That is simple to see from the formula comprising factorials. As an illustration, C(12, 7)  =  C(12, 5). Take whichever one is simpler to find. Is it simpler to find C(100, 2) or C(100, 98)? On calculator it does not make much difference, by hand it does.

Finding Combinations by Hand:

By hand, we can plug the values for n and r to the expression comprising factorials and then simplify the ratio of factorials.

To simplify the ratio, we wish the bigger amount of terms to divide out. For illustration, if you require to find C(12, 5), we could as well find C(12,7). Either way, you are going to have a 12! in numerator and both a 7! and 5! in the denominator. We would instead divide out 7! than 5!, as it leaves you less to work with. Therefore, pick whichever r value is smaller, and then work with the combination.

nCr = (first r factors of n!)/(last r factors of n!)

This turns out the last r factors of n! is really just r!.

Finding Combinations with the Calculator:

There is a permutation function on calculator. On TI-82 and TI-83, it is found beneath the Math menu, the Probability Submenu and then choice 3. This is shown as nCr. Enter the value for n first, then the function, and ultimately the value for r.

Illustrations of Combinations:

We experienced the combinations with Pascal's Triangle; however there are other places they take place.

The old Illinois Lottery had 54 balls, of such 54 balls, six are selected. None of the six can be repeated and the order of six is not significant. That makes it a combination of: C(54,6) = 25,827,165.

It was told that on January 17, 1998, the Illinois Lottery will be changing to the 48 balls, six of which are selected. Now, the number of possibilities will be C(48, 6) = 12,271,512

How many 5 card poker hands are there with the 3 clubs and 2 diamonds? Well, there is no repetition of cards in hand, and the order does not matter, therefore we have a combination again. As there are 13 clubs and we wish 3 of them, there are C(13, 3) = 286 ways to obtain the 3 clubs. As there are 13 diamonds and we wish 2 of them, there are C(13, 2) = 78 ways to obtain the 2 diamonds. As we wish them both to take place at similar time, we utilize the fundamental counting principle and multiply 286 and 78 altogether to get 22,308 possible hands.

Difference between Permutations and Combinations:

The distinguishing character between Permutations and Combinations is not whether or not there is repetition. Neither one permits repetition. The difference among the two is whether or not order is significant. When we have a problem where you can repeat objects, then you should use the Fundamental Counting Principle, you cannot utilize Permutations or Combinations.

Distinguishable Permutations:

Let consider all the permutations of letters in the word BOB.

As there are three letters, there must be 3! = 6 different permutations. Such permutations are BOB, BBO, OBB, OBB, BBO, and BOB. Now, as there are six permutations, a few of them are indistinguishable from one other. When we look at the permutations which are distinguishable, you just have three BOB, OBB and BBO.

To find out the number of distinguishable permutations, take the total number of letters factorial divide by frequency of each and every letter factorial.

152_distinguishable permutations.jpg

Here n1 + n2 + ... + nk = N

Mainly, the little n's are the frequencies of each distinct (distinguishable) letter. Big N is the total number of letters.

Illustration of distinguishable permutations:

Find out the number of distinguishable permutations of letters in the word MISSISSIPPI

Here are the frequencies of the letters. M=1, I=4, S=4, P=2 for a total of 11 letters. Be sure we put parentheses around the denominator and hence we end up dividing by each of the factorials.

11! / (1! * 4! * 4! * 2!) = 11! / (1 * 24 * 24 * 2) = 34,650.

We might want to do some simplification by hand first. Whenever you simplify the ratio of factorials, we get that, there are 34,650 distinguishable permutations in the word MISSISSIPPI. We don't want to list them out, but it is better than listing out all 39,916,800 permutations of the 11 letters in the MISSISSIPPI.

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