Question

If 5 P(4, n) = 6. P (5, n − 1), find n ?

Answer 1

5 P(4, n) = 6. P (5, n − 1)
5 ⁴P_n = 6⁵P_n

-1\[\Rightarrow 5 \times \frac{4!}{\left( 4 - n \right)!} = 6 \times \frac{5!}{\left( 5 - n + 1 \right)!}\]
\[ \Rightarrow 5 \times \frac{\left( 6 - n \right)!}{\left( 4 - n \right)!} = 6 \times \frac{5!}{4!}\]
\[ \Rightarrow 5 \times \frac{\left( 6 - n \right)\left( 6 - n - 1 \right)\left( 6 - n - 2 \right)!}{\left( 4 - n \right)} = 6 \times \frac{5 \times 4!}{4!}\]
\[ \Rightarrow 5 \times \frac{\left( 6 - n \right)\left( 5 - n \right)\left( 4 - n \right)!}{\left( 4 - n \right)} = 6 \times 5\]
\[ \Rightarrow \left( 6 - n \right)\left( 5 - n \right) = 6\]
\[ \Rightarrow \left( 6 - n \right)\left( 5 - n \right) = 3 \times 2\]
\[\text{On comparing the LHS and the RHS, we get}: \]
\[ \Rightarrow 6 - n = 3\]
\[ \Rightarrow n = 3\]