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(52776) About Me  (Show Source):
You can put this solution on YOUR website!
.

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


nCr = %28n%2A%28n-1%29%2A%28n-2%29%2A%28n-r%2B1%29%29%2Fr%21.


So the ratio  nPr/nCr is  r!, and it is equal to  3024%2F126 = 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) About Me  (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:

nCr+=+nPr%2Fr%21

So

r%21+=+nPr%2FnCr+=+3024%2F126+=+24

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.

3024+=+%282%5E4%29%283%5E3%29%287%29

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