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Addition Principle of Counting | Multiplication Principle

Here you will learn addition principle of counting and multiplication principle in permutation and combination with example.

Let’s begin –

If an event A can occur in ‘m’ different ways and another event B can occur in ‘n’ different ways, then the total number of different ways of-

Multiplication Principle of Counting

Simultaneous occurrences of both events in a definite order is m×n. This can be extended to any number of events.

Example : There are 15 IITs in India and let each IIT has 10 branches, then the IITJEE topper can select the IIT and branch in 15×10 = 150 number of ways

Addition Principle of Counting

Happening exactly one of the events is m + n.

Example : There are 15 IITs & 20 NITs in India, then a student who cleared both IITJEE & AIEEE exams can select an institute in (15 + 20) = 35 number of ways.

Factorial Notations

(i)  A useful notation: n! (factorial n) = n.(n – 1).(n – 2)……….3.2.1; n! = n.(n – 1)! where n N

(ii) 0! = 1! = 1

(iii) Factorial of negative integers are not defined

(iv) (2n)! = 2n.n![1.3.5.7……….(2n -1)]

Formation of groups

(a)  (i) The number of ways in which (m + n) different things can be divided into two groups such that one of them contains m things and other has n things, is (m+n)!m!n! (mn).

(ii) If m = n, it means the groups are equal & in this case the number of divisions is (2n)!n!n!2!

As in any one ways it is possible to interchange the two groups without obtaining a new distribution.

(iii) If 2n things are to be divided equally between two persons then the number of ways: (2n)!n!n!2!×2!

(b)  (i) Number of ways in which (m + n + p) different things can be divided into three groups containing m, n & p things respectively is : (m+n+p)!m!n!p!(mnp).

(ii)  If m = n = p then the number of groups = (3n)!n!n!n!3!.

(iii)  If 3n things are to be divided equally among three people then the number of ways in which it can be done is (3n)!(n!)3.

(c)  In general, the number of ways of dividing n distinct objects into x groups containing p objects each and m groups containing q objects each is equal to n!(x+m)!(p!)x(q!)mx!m!.

Example : In how many ways can 15 student be divided into 3 groups of 5 students each such that 2 particular students are always together? Also find the number of ways if these groups are to be sent to three different colleges.

Solution : First pen can be put in 6 ways.

Here first we separate those two particular students and make 3 groups of 5, 5 and 3 of the remaining 13 so that these two particular students always go with the group of 3 students.

   Number of ways = 13!5!5!3!.12!

Now these groups are to be sent to three different colleges, total number of ways = 13!5!5!3!12!.3!

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