Question:
If a denotes the number of permutations of (x 2) things taken all a time, b the number of permutations of x things taken 11 at a time and c the number of permutations of x-11 things taken all at a time such that a=182bc, find the value of x.
Answer:
Given, a denotes the number of permutations of (x + 2) things taken all a time
=> a = x+2Px+2 = (x + 2)!
b the number of permutations of x things taken 11 at a time
=> b = xP11 = x!/(x - 11)!
and c the number of permutations of x-11 things taken all at a time
=> c = x-11Px-11 = (x - 11)!
Again given, a = 182bc
=> (x + 2)! = 182 * {x!/(x - 11)!} * (x - 11)!
=> (x + 2)! = 182 * x!
=> (x + 2) * (x + 1) * x! = 182 * x!
=> (x + 2) * (x + 1) = 182
=> x2 + 3x + 2 = 182
=> x2 + 3x + 2 - 182 = 0
=> x2 + 3x - 180 = 0
=> (x = 12)*(x + 15) = 0
=> x = 12, -15
So, the value of x are 12 and -15
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