A person invites a party of $10$ friends at dinner and place them $5$ at one round table, $5$ at the other round table. $4$ at one round table and $6$ at other round table.The find the ratio, of number of circular permutations in case (i) to case (ii).~

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Solution

In first case the no of permutation is $^{10} C_{5} \times 4! \times 4!$
in second case the no of permutation is $^{10} C_{4} \times 3! \times 5!$
Ratio $= \frac{^{10} C_{5} \times 4! \times 4!}{^{10} C_{4} \times 3! \times 5!} = \frac{4 \times 6}{5 \times 5} = \frac{24}{25}$

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A person invites a party of 10 friends at dinner and place them 5 at one round table, 5 at the other round table.4 at one round table and 6 at other round table.The find the ratio, of number of circular permutations in case (i) to case (ii).~

In first case the no of permutation is 10C5×4!×4!in second case the no of permutation is 10C4×3!×5!Ratio=10C5×4!×4!10C4×3!×5!=4×65×5=2425

A person invites a party of $10$ friends at dinner and place them $5$ at one round table, $5$ at the other round table. $4$ at one round table and $6$ at other round table.The find the ratio, of number of circular permutations in case (i) to case (ii).~

In first case the no of permutation is $^{10} C_{5} \times 4! \times 4!$
in second case the no of permutation is $^{10} C_{4} \times 3! \times 5!$
Ratio $= \frac{^{10} C_{5} \times 4! \times 4!}{^{10} C_{4} \times 3! \times 5!} = \frac{4 \times 6}{5 \times 5} = \frac{24}{25}$