Question

If number of circular permutations of 10 distinct things taken 5 at a time is \(m\) and number of linear permutations of 9 distinct things taken 4 at a time is \(n\), then \(m:n=\)

• A. 1:2
• B. 2:1
• C. 2:3
• D. 3:2

Hint:

Circular arrangement reduces one position because rotation does not create new arrangements.

Answer 1

The Correct Option is B

Concept: Circular permutation formula: \[ {}^nP_r \div r \] or \[ \binom nr(r-1)! \] Linear permutation: \[ {}^nP_r=\frac{n!}{(n-r)!} \]

Step 1: Find circular permutations.
Choose 5 objects from 10. \[ {}^{10}C_5=252 \] Arrange circularly. \[ (5-1)!=24 \] Thus \[ m=252\times24 \] \[ =6048 \]

Step 2: Find linear permutations.
\[ n={}^{9}P_4 \] \[ =\frac{9!}{5!} \] \[ =9\times8\times7\times6 \] \[ =3024 \]

Step 3: Find ratio.
\[ m:n \] \[ 6048:3024 \] \[ 2:1 \] Thus \[ \boxed{2:1} \]