R S Aggarwal Solutions for Exercise 3: EXERCISE 5C

If ${}^{n}P_4:{}^{n}P_5=1:2,$ find $n$.

If ${}^{n-1}{P}_3:{}^{n+1}P_3=5:12,$ find $n$.

If ${}^{2n-1}{P}_n:{}^{2n+1}P_{n-1}=22:7,$ find $n$

If ${}^{2n+1}{P}_{n-1}{:}^{2n-1}{P}_n=3:5,$ find $n$.

If ${}^{11}{P}_r={}^{12}P_{r-1},$ find $r$.

If ${}^{15}P_{r-1}:{}^{16}P_{r-2}=3:4,$ find $r$.

Prove that ${}^{n}P_n=2^n{P}_{n-2}$.

Find the number of permutations of $7$ objects taken $3$ at a time.