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  • If a denotes the number of permutations of x+2 things taken all at a time, b the number of permutations of x things taken 11 at a time and c the number of permutations of x

If a denotes the number of permutations of x+2 things taken all at 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=182 b c, then the value of x is
 

Option: 1

15


Option: 2

12


Option: 3

10

 


Option: 4

18


Answers (1)

\begin{aligned} & { }^{x+2} P_{x+2}=a \Rightarrow a=(x+2) ! \\ \\& { }^x P_{11}=b \Rightarrow b=\frac{x !}{(x-11) !} \end{aligned}

and  { }^{x-11} P_{x-11}=c \Rightarrow c=(x-11) !

\begin{array}{lc} \therefore & a=182 b c \\ \\& (x+2) !=182 \frac{x !}{(x-11) !}(x-11) ! \\ \\\Rightarrow & (x+2)(x+1)=182=14 \times 13 \\ \\\therefore & x+1=13 \\ \\\therefore & x=12 \end{array}

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Kshitij

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