Then the permutations of these objects, taking two at a time, are xy, yx, yz, zy, xz, zx = Total 6 Permutations. So the number of ways in which a boy does not get all the prizes = 43 – 4 = 60Įach of the arrangements which can be made by taking some or all of a number of things is called a permutation.įor example, if there are three objects namely x,y, and z, So, the number of ways in which a boy gets all the 3 prizes = 4 III) Since any one of the 4 boys may get all the prizes. Similarly second to any one of the 4 boys, and third as well to any one of the 4 boys = 4 X 4 X 4 = 43 = 64 Note :- This is same as Arrangement of 4 boys taken 3 at a time in a way 4P3 = 4!/1! = 4! = 24 (More on this under the Head of Permutations later) Hence total number of ways are 4 X 3 X 2 = 24 Similarly third prize can be given away to any of the remaining 2 boys The second prize can be given away to any of the remaining 3 boys because the boy who got the first prize cannot receive second prize. I) The first prize can be given away to any of the 4 boys, hence there are 4 ways to distribute first prize. Hence total number of ways = 7 X 6 X 5 X 4 X 3 = 2520Įxample 4 :- In how many ways can 3 prizes be distributed among 4 boys, when In the same way d has 5, e has 4, and f has 3 way. c can leave the cabin in 6 ways, since he can not leave at where b left. Thus the total number of ways in which each of the five persons can leave the cabin at any of the seven floors is Similarly each of c,d,e,f also has 7 options. I) b can leave the cabin at any of the seven floors. Find the total number of ways in which each of the five persons can leave the cabin Suppose each of them can leave the cabin independently at any floor beginning with the first. Number of ways going out of the room = 5 (He/She cannot go from the same door)īy Fundamental Principle of Multiplication-> Coming in X Going out = 6 X 5 = 30.Įxample 3 :- Five persons entered the lift cabin on the ground floor of an 8 floor house. In how many ways can a person enter the room through one door and come out through a different
Number of ways of awarding one of the three scholarships = 3 + 5 + 4 = 12-Įxample 2 :- A room has 6 doors. In How many ways one of these scholarships be awarded? Number of ways of awarding three scholarshipsV= 3 X 5 X 4 = 60. In How many ways can these scholarships be awarded?Ĭlearly classical scholarship can be awarded to anyone of the 3 candidates, similarly mathematical and natural science scholarship can be awarded in 5 and 4 ways respectively. Hence the teacher can make the selection of a student in 18 ways.Įxamples 1 :- There are 3 candidates for a classical, 5 for a mathematical, and 4 for a natural science scholarship. Therefore, by fundamental principle of addition either of the two jobs can be performed in (8 + 10) ways. The first of these can be performed in 8 ways and the second in 10 ways. Here the teacher has to choose either a girl OR a boy (Only 1 student)įor selecting a boy she has 8 options/ways OR that for a girl 10 options/ways. In how many ways can she make her selection? "If there are two jobs such that they can be performed independently in ‘m’ and ‘n’ ways respectively, then either of the two jobs can be performed in (m + n) ways."Įxample :- In her class of 10 girls and 8 boys, the teacher has to select either a girl OR a boy. Remark :- The above principle can be extended for any finite number of jobs. Here the teacher has to choose the pair of a girl AND a boyįor selecting a boy she has 8 options/ways AND that for a girl 10 options/waysįor 1st boy - any one of the 10 girls - 10 waysįor 2nd boy - any one of the 10 girls - 10 waysįor 3rd boy - any one of the 10 girls - 10 waysįor 8th boy - any one of the 10 girls - 10 ways "If there are two jobs such that one of them can be completed in ‘m’ ways, and another one in ‘n’ ways then the two jobs in succession can be done in ‘m X n’ ways."Įxample :- In her class of 10 girls and 8 boys, the teacher has to select 1 girl AND 1 boy. In fact these two principles form the base of Permutations and Combinations. These two principles will enable us to understand Permutations and Combinations. principle of addition and principle of multiplication. Here we shall discuss two fundamental principles viz.
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