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

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

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L.H.S. = `n_(P_r)​=(n!)/((n−r)!)` ..(1)
​ R.H.S. =` (n−1)_(P_r)​+r(n−1)_(P_(r−1))`
​ ​=`((n−1)!​)/((n−r−1)!)+((n−1)!​)/((n−r)!)`
= `((n−1)!​)/((n−r−1)!)[1+r/(n−r)​]`
=`(n.(n−1)!​)/[ (n−1).(n−r−1)!]`
=`(n!)/((n−r)!)`​ ..(2)
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RD SHARMA-PERMUTATIONS-Solved Examples And Exercises
  1. If ^0P5+5^9P4=^(10)Pr , find the value of rdot

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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  20. If P(9, r)=3024 ,\ find rdot

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