1. 1. What is permutation Formulas permutation Formula Derivation Cards types Use of permutation in civil engineering
2. 2. In probability and statistics, the permutation is defined as the : Each of several possible ways in which a set number of things can be arranged
3. 3. Permutations Repetition allowed Repetition is not allowed
4. 4. Another defination is : An arrangement of n number of objects (distinct) taken r at a time in specific order is known as permutation denoted by ( nP )
5. 5. FORMULA: nP= n! (For the repeating permutation)
6. 6. FORMULA: nP = n! (For non repeating permutation) (n-r)!
7. 7. Lets derive the formula for repeating and non repeating Permutations : Suppose that we are given n distinct objects and we wish to arrange r of these objects in a line . Any of the objects can fill up the first place . when the first place has been filled by one of the n ways then there are (n-1) objects left for filling second place .
8. 8. Hence by multiplication rule there will be nx(n-1) ways of filling up the first two places. There will be (n-2) ways to fill up the 3rd place . Hence the first three places can be filled up by nx(n-1)x(n-2) ways. Continuiing this way we get .
9. 9. The last which is the r th place can be filled by the following: n x(n-1)x(n-2) r factors Note that the first factor is (n-1+1)=n Second factor is (n-1)=(n-2+1) etc Its clear that the r th factor is (n-r+1) For each place to be filled. -1 for first place -2 for second place And so on. For each progressive place to be filled adding +1 each time.
10. 10. Hence the number of permutations of n objects taken r at a time are : nP = n(n-1) (n-r+1) In particular if all n items are to be permutted then: nP = n(n-1)(n-{n-2}+1)(n-{n-1}+1)(n-n+1) nP = n(n-1) 3.2.1 Consider n=10 (n-n+1) =(10-10+1) = 1 (n-{n-1}+1) =(10-{10-1}+1) = 2 (n-{n-2}+1) =(10-{10-2}+1) = 3 And so on.
11. 11. From previous slide nP = n(n-1) (n-r+1) For r=n for last term nP = n(n-1)(n-{n-2}+1)(n-{n-1}+1)(n-n+1) nP= n(n-1) 3.2.1 nP= n! (For the repeting permutation) (! = factorial)
12. 12. Now for the non repeating permutation multiplying and dividing the expression for nP by (n-r)(n-r-1)3.2.1 : nP = n(n-1) (n-r+1) (n-r)(n-r-1)3.2.1 (n-r)(n-r-1)3.2.1 nP = n! (For non repeating permutation) (n-r)!
13. 13. nP= n! (For the repeating permutation) nP = n! (For non repeating permutation) (n-r)!
14. 14. DECK OF 52 CARDS 13 OF EACH SUIT
15. 15. FOR EXAMPLE : During the construction of a mosque a builder comes for asking the arab of Dubai that where do you want these 7 doors to be built and in what order .The Arab thinks but gets confused and asks that if you answer my question I will give you 1 lakh dollars.Builder says yeah ask . So arab says how many total ways are there that I can arrange these 7 doors distinctly ?
16. 16. Meeting the confused Arab
17. 17. SUPPOSE THAT YOU GUYS ARE THE BUILDER WHO HAS GOT THE OPPURTUNITY TO EARN 1 LAKH DOLLARS FROM ARAB WHAT WILL BE YOUR ANSWER ???
18. 18. Solution: Total doors of mosque = n =7 Arrangements of doors = r =7 For non repeating permutations: nP = n! (n-r)! nP = 7! = 5040 ways in which arrangements can be done (7-7)!
19. 19. Always do something different because history is made by those who dont follow rules .. Bring the change ,create new standards.
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