Question

If a denotes the number of permutations of x+2 things taken all at a time, b denotes the number of permutations of x things taken 11 at a time and c is the number of permutations of x-11 things taken all at a time such that a = 182bc, then the value of x is:
1)15
2)12
3)10
4)18

Answer 1

Hint: This question requires one to know only one formula, i.e., \[^n{P_r}\]
\[{ \Rightarrow ^n}{P_r} = \dfrac{{n!}}{{(n - r)!}}\], where n = the total things on which permutations are performed and
 r = denotes the number of things taken at a time. We will first write the expressions of permutations for the questions and then simplify their values to get to the final answer of the question.

Complete step-by-step answer:
Let’s start the question by calculating the value of a
Now, the condition for a was the arrangement of x+2 things taken x+2 times, thus a is given by
\[ \Rightarrow a{ = ^{x + 2}}{P_{x + 2}}\]
\[ \Rightarrow a = \dfrac{{(x + 2)!}}{{((x + 2) - (x + 2))!}}\]
\[ \Rightarrow a = (x + 2)! - - - - (i)\]
Now, let’s find the value of b
For b the condition was the number of arrangements of x things taken 11 at time, thus b is given by
\[ \Rightarrow b{ = ^x}{P_{11}}\]
\[ \Rightarrow b = \dfrac{{(x)!}}{{(x - 11)!}} - - - - (ii)\]
And finally, let’s calculate the value of c
For, c the condition was the arrangement of x-11 things taken x-11 times, thus c is given by
\[ \Rightarrow c{ = ^{x - 11}}{P_{x - 11}}\]
\[ \Rightarrow c = \dfrac{{(x - 11)!}}{{((x - 11) - (x - 11))!}}\]
\[ \Rightarrow c = (x - 11)! - - - - (iii)\]
The condition given is
\[ \Rightarrow a = 182bc\]
Now, using \[(i)\],\[(ii)\]and \[(iii)\]we get
\[ \Rightarrow (x + 2)! = 182 \times \dfrac{{(x)!}}{{(x - 11)!}} \times (x - 11)!\]
\[ \Rightarrow (x + 2)(x + 1) = 182\]
Here, x+2 and x+1 are consecutive , so, let’s try to break 182 as a product of 2 consecutive numbers or \[182 = 14 \times 13\]
\[ \Rightarrow (x + 2)(x + 1) = 13 \times 14\]
Equating \[(x + 1)\]to \[13\]( because clearly \[(x + 1)\]is smaller than \[(x + 2)\]), we get
\[ \Rightarrow x + 1 = 13\]
\[ \Rightarrow x = 12\]
Thus, option(2) is the correct answer.
So, the correct answer is “Option 2”.

Note: We should know about the formulae of permutations and combinations while solving such questions. We should take care while canceling the common factors in numerator and denominator. verify the calculations once so as to be sure of the final answer.