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The Basic Counting Principle-Statistic for Management Notes

In: Business and Management

Submitted By suji1234
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The Basic Counting Principle
When there are m ways to do one thing, and n ways to do another, then there are m×n ways of doing both.
123
132
213
231
321
312
These are the possible three digit no.s

3
2
3
1
1
2
2
3
1
3
2
1
In how many ways can three digit number be formed from the numbers 1,2,3 without repeating the digits within the numbers.
1

2

3

3 options * 2 options * 1option=6 ways

Find out the possible number of arrangements below:
S1 T1
S2 T2
S3
Here n is 3, r is 2 For principle of counting we must have same number of options. In above if S3 and T2 is not allowed, principle of counting doesn’t work
If n=5 and r=3 {A,B,C,D,E}
How many different ways can we arrange of taking 3 letters at a time?
5 *4*3= 60 ways

This is permutation of n different thing taken r at a time
60=(5*4*3*2*1)/(2*1) = 5!/2!=5!(5-3)!=n!/(n-r)!
We are talking about linear arrangement not the circular one here nPr= filling r places by n different thing n=5 {A,B,C,D,E} r=3 {A,B,C}, {A,B,D}, {A,C,D}, {A,C,E}…………..
[Note: Arrangement is related to permutation. If we are considered about place or position it is permutation question. Selecting is related to permutation. If we are not considered about place or position it is combination question.]

nCr, C(n,r)= n!/[(n-r)!r!] n=5 {A,B,C,D,E} r=3 {A,B,C}, {A,B,D}, {A,B,E}, {A,C,D}, {A,C,E},{A,D,E}, {B,C,D},{B,C,E}
3 are selected out of 5 . Above are possible combinations.
Permutation: {A,B,C} and {A,C,B} are different because here, in {A,B,C} second place we have B but in {A,C,B} second place we have C
Combination: {A,B,C} and {A,C,B} are same because in both of them letters A,B,C are selected.
{A,B,C} n=3 and r=3 then, with repetition no of arrangements = 3*3*3=27 ways but without repeatation number of ways = 3*2*1= 6 ways…...

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