Question

If the number of circular permutations of 9 distinct things taken 5 at a time is \(n_1\), and the number of linear permutations of 8 distinct things taken 4 at a time is \(n_2\), then what is \(\frac{n_1}{n_2}\)?

• A. \(\frac{5}{9}\)
• B. \(2\)
• C. \(\tfrac{1}{2}\)
• D. \(\frac{9}{5}\)

Hint:

Circular permutations of \(r\) objects from \(n\) distinct items is \(\binom{n}{r}(r-1)!\).
- Linear permutations of \(r\) from \(n\) is \(P(n,r) = \frac{n!}{(n-r)!}\).

Answer 1

The Correct Option is D

Step 1: Formula for circular permutations of 9 distinct items taken 5 at a time.
First choose 5 out of 9, then arrange them in a circle: \[ n_1 = \binom{9}{5} \times (5-1)! = \binom{9}{5}\times 4!. \] \[ \binom{9}{5} = 126, \quad 4! = 24, \quad \Rightarrow n_1 = 126\times 24 = 3024. \] Step 2: Formula for linear permutations of 8 things taken 4 at a time.
\[ n_2 = P(8,4) = 8 \times 7 \times 6 \times 5 = 1680. \] Step 3: The ratio \(\frac{n_1}{n_2}\).
\[ \frac{n_1}{n_2} = \frac{3024}{1680} = \frac{9}{5}. \] Hence \(\boxed{\tfrac{9}{5}}\).