Question

If a denotes the number of permutations of \[\left( {x + 2} \right)\;\]things taken all at a time, b the number of permutations of $ x $ things taken 11 at a time and c the numbers of permutations of $ (x - 11) $ things taken all at a time such that $ a = 182bc, $ then the value of $ x $ is greater than or equal to

Answer 1

Hint: Number of permutations is equal to the number of possible arrangements. Number of permutations of n different things taken at a time is given by $ n! $

Complete step-by-step answer:
Given:
Let $ a $ denotes the number of permutation of $ (x + 2) $ things.
 $ \therefore $ permutation for $ (a) $ $ = (x + 2)! $
Now, let $ b $ denotes the number of permutation of $ x $ things taken $ 11 $ at time
Since number of permutations of $ n $ different things taken $ r $ at a time is given by $ ^n{P_r} = \dfrac{{n!}}{{(n - r)!}} $
We use the above formula to write $ \Rightarrow b = \dfrac{{x!}}{{(x - 11)!}} $
It is given that, $ a = 182bc $ . . . (1)
Let $ c $ denotes the number of permutations of $ (x - 11) $ things.
 $ \Rightarrow c = (x - 11)! $
Now, substituting the values of a, b, and c in equation (1), we get
 $ (x + 2)! = 182 \times \dfrac{{x!}}{{(x - 11)!}} \times (x - 11)! $
We know that $ n! = n(n - 1)(n - 2)....3 \times 2 \times 1 $
Also by using properties of factorial we can write $ n! = n(n - 1)! $
By using these formulae, we can simplify the above equation as
 $ (x + 2) \times (x + 1) \times x! = 182x! $
Cancelling the common term from both the sides, we get
 $ (x + 2) \times (x + 1) = 182 $
 $ \Rightarrow x(x + 1) + 2(x + 1) = 182 $
 $ \Rightarrow {x^2} + x + 2x + 2 = 182 $
 $ \Rightarrow {x^2} + 3x + 2 - 182 = 0 $
 $ \Rightarrow {x^2} + 3x - 180 = 0 $
This is a quadratic equation. We will solve it using factorization method
 $ \Rightarrow {x^2} + 15x - 12x - 180 = 0 $ (using the method of splitting the middle terms)
 $ \Rightarrow x(x + 15) - 12(x + 15) = 0 $
 $ $ $ \therefore (x + 12)(x + 15) = 0 $
 $ \Rightarrow x - 12 = 0 $ or $ x + 15 = 0 $
 $ \Rightarrow x = 12 $ or $ x = - 15 $
But $ x $ cannot be negative as it represents the number of things. Therefore we will consider the positive value of $ x $
 $ \therefore $ The value of \[x\]is \[12.\]

Note: Formula of permutation of $ n $ things changes if all the things are not different. So you have to be careful while using the formula used above. First read the question carefully to make sure if given objects are all different or not.