Question

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 bc, find the value of x.

Answer 1

a = ^x+2P_x+2 = (x+2)!
b  = ^xP₁₁ =\[\frac{x!}{(x - 11)!}\]

c= x\[-\]¹¹P_x\[-\]₁₁ =\[(x - 11)!\]

a = 182 bc

\[\Rightarrow\] (x+2)! = 182\[\frac{x!}{\left( x - 11 \right)!}\]\[\times \left( x - 11 \right)!\]

\[\Rightarrow\](x+2)! = 182 (x!)

\[\Rightarrow \frac{\left( x + 2 \right)!}{x!} = 182\]
\[ \Rightarrow \left( x + 2 \right)\left( x + 1 \right) = 182\]
\[ \Rightarrow \left( x + 2 \right)\left( x + 1 \right) = 14 \times 13 \]
\[ \Rightarrow x + 2 = 14\]
\[ \Rightarrow x = 12\]