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In our daily life, we face different problems related to finding the number of ways to set an object or to group and arrange it under certain conditions.Such concepts of namely grouping and arranging are discussed in this chapter.Basically, these are the problems of counting and such problems are studied under the branch of mathematics called Combinatorics.Permutation and Combination are studied under this branch of mathematics.
The arrangements or grouping in which the arrangements of the events or items are certain is known as Permutation.Combinationrefers to the arrangements or grouping in which the order doesn't matter.
The fundamental/basic principle of counting is a simple multiplication process.It helps us deal with the number of ways any work can be done.let us say that we can do a work in say 'm' ways and the no of ways to do another work is 'n'.Then the principle of counting tells us that the no ways to do both works are 'mxn' ways.This rule also complies even if the no. of works is more than two.for a basic concept let us say you have 3 shirts and 4 pants.Then the no. of ways you can wear ur shirt and pant(different dress combination) is given by 3x4 = 12 ways.
Let us suppose, we have four teams involved in a football tournament and we need to find out the number of possibilities for 1st and 2nd place.Then we can use the principle of counting to find it. Let us suppose the teams are A, B, C and D.Then the 1st place can either be team A, B, C and D ie 4 ways.For the second place, there are only three options as one team will be in the 1st place. So 2nd place has 3 ways/possibilities.Let us understand from the table below:
|First Place||Second Place possibilities||Final standing|
|A||B or C or D||1st:A 2nd :B or AC or AD|
|B||C or D or A||BC or BD or BA|
|C||A or B or D||CA or CB or CD|
|D||A or B or C||DA or DB or DC|
From the table above it is clear that the 1st place has 4 choices (A,C,B,D) and 2nd place has 3 remaining choices.We can also see from the table that there are 12 ways the final standings can occur which proves the point of the basic principle of counting. so, we can easily know that the no ways is 4x3 =12 for the 1st and second position from the principle of counting.
Example 1: P, Q, R, and S are four cities. There are 5 roads from P to Q and four routes from R to S. In how many ways can a driver drive from P to S via Q and R?
There are five roots from P to Q, the driver can P to Q in 5 ways. Then he can drive from Q to R in 3 ways and then from R to S in 4 ways. $$\therefore The\; driver\;can\; drive \;from \;P \;to\; S\; in\;5 \;\times\;3\;\times\; 4\;=\;60\;ways.$$
Example2: How many 2 digit numbers can be found using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 if no digit is repeated? Solution:
Two digit numbers have two places, the one's/unit place, and the tens place. For the tens place, we can use any digit among the given except 0 we can use any of the remaining 9 digits or ways. For the one's place, out of the 10 digits, one has already been used in the tens place.so, we can use any 9 of the remaining digits.
$$\therefore The\;two\;digits\;numbers\;(without\;replacement\;of\;digit) \;= \;9\; \times \;9 \;= \;81$$
Note: If repetition is allowed, the total number of two digits numbers formed =9 x 10 = 90
Example 3: How many numbers of three different digits less than 500 can be formed from the integers 1,2,3,4,5,6?
Here the number of given digits = 6. Since the number is less than 500, only 1, 2, 3, 4 can be used for the hundredth place. So, the hundredth place has 4 places. The tens place can be chosen out of 5 digits and the unit place can be chosen in 4 ways.
$$\therefore The\;numbers\;of\;ways\;for\;forming\;3\;different \;digits\;number\;=\;4\;\times\;5\;\times\;4\;=\;80\;ways$$.
Example 4: How many numbers are there between 100 and 1000 such that every digit is either 2 or 9 ?
Any number between 100 and 1000 is a 3-digit number for sure. Now the unit's place has two choices or ways 2 or 9. The same choice is available for both ten's and hundredth's place.
The above examples show the use of the principal of counting to find the number of ways to do a combined work/thing when the number of ways for separate work or thing is given ( i.e. multiplication of the number of ways of completing each work or thing ).
Taken reference from
( Basic mathematics Grade XII and A foundation of Mathematics Volume II and Wikipedia.com )