What Is the Solution for n in the Equation 10Pn = 90?

AI Thread Summary
To solve the equation 10Pn = 90, the factorial relationship is utilized, leading to the equation 40320 = (10 - n)!. By recognizing that 40320 equals 8!, it follows that (10 - n) must equal 8, resulting in n = 2. Additionally, for the student council election problem, each student can vote for multiple positions, leading to a combinatorial calculation of the total ways a ballot can be marked. The discussion highlights the importance of understanding permutations and combinations in solving these types of problems.
blue_soda025
Messages
26
Reaction score
0
What would be the best way to solve for n if 10Pn = 90?
Also, how would you solve this problem:
In a student council election, there are 3 candidates for president, 3 for secretary, and 2 for treasurer. Each student may vote for at least one position. How many ways can a ballot be marked?
Thanks in advance.
 
Physics news on Phys.org
For the first one, use the fact that _{n} P _{k} = \frac{n!}{(n-k)!}
 
I used that and multiplied both sides by (10 - n)!, then divided both sides by 90. Then I got 40320 = (10 - n)!. But that's where I got stuck.
 
Try expressing 40320 as a factorial.
 
I suppose it would be 8! = (10 - n)! then? Still don't know what to do...
 
if 8! = (10-n)!
n has to equal 2.
 
Oh, I see now.. don't know why I didn't before. Thanks!
 
Back
Top