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Prove the following: P(n , n)=2P(n , n-2...

Prove the following: `P(n , n)=2P(n , n-2)`

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We know that,`n_(P_n)​=(n!​)/((n−r)!)`
Hence
`n_(P_n)​=(n!​)/((n−r)!)=(n!)/(0!)​=n!`
Also,`n_(P_n​)=(n!)/((n−(n−2))!)`
​`=(n!​)/((n−n+2)!)=(n!​)/(2!)=(n!)/2`
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RD SHARMA-PERMUTATIONS-Solved Examples And Exercises
  1. If 10(Pr)=5040 , find the value of rdot

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  2. If ^0P5+5^9P4=^(10)Pr , find the value of rdot

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  3. Prove the following: P(n , n)=2P(n , n-2)

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  4. Prove the following: P(n , r)=P(n-1, r)+rdotP(n-1,\ r-1)

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  5. Prove the following: n(Pn)​= n(P(n−1))

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  6. Prove the following: P(n , r)=ndotP(n-1,\ r-1)

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  7. In how many ways three different rings can be worn in four fingers wit...

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  8. How many different signals can be made by 5 flags from 8 flags of d...

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  9. Three men have 4 coats, 5 waist coats and 6 caps. In how many ways c...

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  10. How many numbers lying between 100 and 1000 can be formed with the dig...

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  11. How many four digit numbers are there with distinct digits?

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  12. In how many ways 7 pictures can be hung from 5 picture nails on a wall...

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  13. The number of natural numbers smaller than 10^4 of which all digits ar...

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  14. How many words, with or without meaning, can be formed using all th...

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  15. Ten different letters of an alphabet are given. Words with five letter...

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  16. Evaluate each of the following: \ ^8P3

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  17. Evaluate each of the following: \ ^(10)P4

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  18. Evaluate each of the following: P(6,4)

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  19. If P(5, r)=P(6, r-1),\ fin d\ rdot

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  20. If \ ^n P4=360 ,\ find the value of ndot

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