Permutation Complete Note Class 12 Math Algebra PDF NEB (Permutation And Combination) Theorems And Proofs . MCQ
In this article , you will find the complete solution of Permutation Class 12 Math . The complete Chapter name is Permutation And Combination . This is the second part . The First Part is Uploaded .Basic Principle of Counting All the theorems and proved with solutions and important questions . You can also download the full PDF for read offline. Follow Edubook Ram Kumar Sah for more study materials like this . Share this blogs and comment your feedback .
Permutation
The different arrangements, which can be made out of a given number of things by takin some or all at a time, are called Permutations. P(n, r ) or nPr denotes the number of permutations of In n things taken r at a time .
Theorem 1:
The number of permutations of n different things taken r at a time is given by
Proof:
The number of permutations of n different things taken r at a time is equivalent to the number of ways in which r position can be filled up byu those n things.
The first place can be filled up in n different ways, for any one of the things can be put in this place.
After the filing the first place, we are left with (n-1) things. Now the second place can be filled up in (n-1) different ways. because any one of the (n-1) things can be put in this place.
Similarly, for the third place, we have (n-2) choices and so on. Ultimately to fill up the the rth position, there are n-(r-1)=n-r+1 choices. Then, by basic principle of counting, the number of ways in which r positions can be filled by n different things is n(n-1) (n-2)... (n-r+1).
Therefore, P(n, r) = n(n- 1) (n - 2)... (n - r + 1)
Corollary : The number of permutations of n different things, takern all at a time is given by
Proof : In theorem 1, putting r = n , we get
The number or permutations of a set of n objects taken all at a time when there are p objects are of same kind, q objects are of same kind and all other being different is
Proof:
This theorem can be verified by taking example of the number of permutations in the word "TOMORROW"
In this word, there are 8 letters not all different. There are 3 O's i.e. 3 letters alike, 2R's i.e. two letters a like and all other letters are different. If all the letters are different, the number of permutations would have been 8! . But as there are 3 O's, interchange of 3 O's among them does not produce different words. So, number of permutation is less than 8!. Let the number be x. If the 3 O's also are different, they can be arranged among themselves in 3! ways. From each word, we set 3! words simply be arranging 3 O's . But there are x words. Hence 3!x is the number of permutation obtained by considering the 3 O's also different.
Again, Let us take any one of these 3!x permutations and assume 2R's are different.
So they can be arranged among themselves in 2! ways. Thus one permutatons gives rise to 2! ways. Thus one permutation gives rise to 2! permutations.
Hence, 3!x permutations will give 3! 2!x permutations.
But, at this stage all the letters become dissimilar (different) and the number of permutations of letters taken all at a time = 8!.
Hence, 3!. 2!x = 8!
∴ x=
Generalization of this example stablished the following theorem:
Exercise Of Permutation Class 12 Mathematics
1. (a) Define the permutation with an example.
(b) If P(5, r) = 2P(6, r-1), find r.
2. IF 10Pr =5400, find the value of r.
3. Find the number of permutations of 5 different object taken 3 at time.
4. If 3 persons enter a bus in which there are ten vacant seats, in how many wats they can sit?
5. How many permutations are there using the letters of the following words:
(a) BIRATNAGAR (b) MAHENDRANAGAR (c) MATHEMATICS (d) ACCELERATION
6. In how many wats can four boys and three girls be seated in a row containing seven seats if:
(a) they sit anywhere ?
(b) The boy and girls must sit alternatively?
7. In how many ways can ten people be seated in a row of eight seats if three people consist on sitting next to each other ?
8. Seven athletes are participating in a race. In how many ways can the first three prizes be won ?
9. In how many wats can seven different coloured beads be made into a bracelet?
10. Of the numbers formed by using all the figures 1, 2, 3, 4, 5 only once , how many are even?
11. How many numbers between 4000 and 5000 can be formed with digits 2, 3, 4, 5, 6, 7?
12. How many words can be formed with the letters of the word 'NUMBER' so that vowels occupy odd place?
13. in How many wats 6 boys and 3 girls can be seated in a row o that no two girls are together?
14. How many words can be formed from the letters of the word ' DAUGHTER' so that :
(a) the vowels always come together?
(b) the vowels never come together?
15. How many arrangement can be made with the letters of the word ' MATHEMATCS' ? In how many of them vowels are together?
16. Three are 4 blue balls, 2 red balls and 3 green balls. In how many ways can the y be arranged in a row?
17. How many different words can be formed by using all the letters of the word 'TUESDAY' . How many of these arrangements do not begin with 'T' ? How many begin with 'T' and do not end with 'Y' ?
18. How many numbers of 3 digits can be formed with the digits 1, 2, 3, 4 and 5, when digits may ne repeated?
Solve Questions and Answer of Permutation Class 12 Mathematic [ PDF ]
In this article you will able to solve almost all question of permutation chapter of mathematics. Here is the PDF to learn more question on this topics. You can find Past questions, Most important questions of permutation also solutions of each problem .
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