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

Step 1: Recall the circular-permutation count.
The number of circular arrangements of $r$ objects chosen from $n$ distinct objects is $\binom{n}{r}(r-1)!$.
Step 2: Compute m.
Choose $5$ from $10$: $\binom{10}{5}=252$. Arrange the $5$ in a circle: $(5-1)!=24$. So $m=252\times 24=6048$.
Step 3: Recall the linear-permutation count.
The number of linear arrangements of $r$ objects from $n$ is ${}^{n}P_r=\dfrac{n!}{(n-r)!}$.
Step 4: Compute n.
$n={}^{9}P_4=9\times 8\times 7\times 6=3024$.
Step 5: Form the ratio.
$m:n=6048:3024$.
Step 6: Simplify.
Dividing both by $3024$ gives $2:1$, option (2).
\[ \boxed{m:n=2:1} \]