SOLUTION: given nPr=3024 and nCr=126 find n and r

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Question 1147834: given nPr=3024 and nCr=126 find n and r
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52777)   (Show Source): You can put this solution on YOUR website!
.

nPr = n*(n-1)*(n-2)*(n-r+1).


nCr = .


So the ratio  nPr/nCr is  r!, and it is equal to   = 24.


It implies  r = 4.


Then  nPr = n*(n-1)*(n-2)*(n-3) = 3024.


I made a short table using MS Excel in my computer


    n       n*(n-1)*(n-2)*(n-3)

    4		24
    5		120
    6		360
    7		840
    8		1680
    9		3024
   10		5040


which shows that  n= 9.


ANSWER.  r = 4;  n = 9.

Solved.




Answer by greenestamps(13198)   (Show Source): You can put this solution on YOUR website!


The difference between nPr and nCr is a factor of r! in the denominator for nCr:



So



So r is 4.

So nPr = 3024 is the product of four consecutive integers. Find the integers by looking at the prime factorization of 3024 and grouping the prime factors to form those four integers.



Clearly one of the four integers is 7.

And there is no factor of 5, so neither 5 nor 10 can be one of the integers.

Therefore the four integers are 6, 7, 8, and 9; so n is 9.

ANSWER: n=9; r=4
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