The symbol ! means factorial, which is defined recursively as
n! =n*(n-1)!
0! =1
Then, 9!
=9*8!
=9*8*7!
=9*8*7*6!
=9*8*7*6*5!
=9*8*7*6*5*4!
=9*8*7*6*5*4*3!
=9*8*7*6*5*4*3*2!
=9*8*7*6*5*4*3*2*1!
=9*8*7*6*5*4*3*2*1*0!
=9*8*7*6*5*4*3*2*1*1
=362880
As seen, the factorial of a certain integer is basically multiplying all integers between 1 and that integer.
""_nP_k is defined as (n!)/((n-k)!). This is used to determine the number of ways you can choose k items from n items, where the order is important. ""_25P_3 is then (25!)/((25-3)!)=(25!)/(22!). Expanding this out using the process similar to the previous problem, we get (25*24*23*22*21*20*19*18*17*16*15*14*13*12*11*10*9*8*7*6*5*4*3*2*1)/(22*21*20*19*18*17*16*15*14*13*12*11*10*9*8*7*6*5*4*3*2*1).
Notice that we can cancel some of these factors: (25*24*23*cancel(22)*cancel(21)*cancel(20)*cancel(19)*cancel(18)*cancel(17)*cancel(16)*cancel(15)*cancel(14)*cancel(13)*cancel(12)*cancel(11)*cancel(10)*cancel(9)*cancel(8)*cancel(7)*cancel(6)*cancel(5)*cancel(4)*cancel(3)*cancel(2)*cancel(1))/(cancel(22)*cancel(21)*cancel(20)*cancel(19)*cancel(18)*cancel(17)*cancel(16)*cancel(15)*cancel(14)*cancel(13)*cancel(12)*cancel(11)*cancel(10)*cancel(9)*cancel(8)*cancel(7)*cancel(6)*cancel(5)*cancel(4)*cancel(3)*cancel(2)*cancel(1)). We are left with 25*24*23, which is equal to 13800.
