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 formula.
Circular arrangements of $r$ chosen from $n$ distinct objects number $\binom{n}{r}(r-1)!$.
Step 2: Find m.
$\binom{10}{5}=252$ and $(5-1)!=24$, so $m=252\times 24=6048$.
Step 3: Recall the linear formula.
Linear arrangements number ${}^{n}P_r=\dfrac{n!}{(n-r)!}$.
Step 4: Find n.
$n={}^{9}P_4=9\times 8\times 7\times 6=3024$.
Step 5: Write the ratio.
$m:n=6048:3024$.
Step 6: Reduce it.
Both divide by $3024$, giving $2:1$, option (2).
\[ \boxed{m:n=2:1} \]