# A Problem with Permutations

Hey guys , Could anyone here tell me the easiest way to solve for n , nP7 = 604800 , the traditional way (I'm currently using) is to divide 604800 by 10 and then 9 and so on until I get 1 as a result of that division , The problem is this way isn't helpful with all permutations I have in my study , some times it gets tricky .

Mentallic
Homework Helper
Notice that 604,800 is divisible by 100, which means that 10 must be one of the numbers in the factors that multiply to give $^nP_7$. This is because we only have one prime factor of 5 which can couple with a prime factor of 2 in various ways, such as from 4 or 6 or 8, and that would give a factor of 10, so the other factor of 10 must be from 10 itself, or even 15 (the factor of 5 coupled with another 2 elsewhere). However, 15*14*...*9 is composed of 7 factors that are all mostly larger than 10, so the result is going to be larger than 107 = 10,000,000 which is much too large. So we only have a few to test, starting at 10*9*...*4 and working our way upwards from there. We actually find that n=10 gives the answer, but if for argument's sake we had to solve, say,

$^nP_7=3,991,680$

Then we know that n=15 is again too large, but now it's also not divisible by 100 any more, which means we're missing a prime factor of 5, and how can that happen? Only with
12*11*...*6
13*12*...*7
14*13*...*8
n being one larger or smaller than these possibilities and we'd end up with the factor of 5 in 5 or 15 and the result would be divisible by 100. So given just a few values to test, the result should be quick to find.

Notice that 604,800 is divisible by 100, which means that 10 must be one of the numbers in the factors that multiply to give $^nP_7$. This is because we only have one prime factor of 5 which can couple with a prime factor of 2 in various ways, such as from 4 or 6 or 8, and that would give a factor of 10, so the other factor of 10 must be from 10 itself, or even 15 (the factor of 5 coupled with another 2 elsewhere). However, 15*14*...*9 is composed of 7 factors that are all mostly larger than 10, so the result is going to be larger than 107 = 10,000,000 which is much too large. So we only have a few to test, starting at 10*9*...*4 and working our way upwards from there. We actually find that n=10 gives the answer, but if for argument's sake we had to solve, say,

$^nP_7=3,991,680$

Then we know that n=15 is again too large, but now it's also not divisible by 100 any more, which means we're missing a prime factor of 5, and how can that happen? Only with
12*11*...*6
13*12*...*7
14*13*...*8
n being one larger or smaller than these possibilities and we'd end up with the factor of 5 in 5 or 15 and the result would be divisible by 100. So given just a few values to test, the result should be quick to find.

Mentallic
Homework Helper