Aptitude Discussion

Q. |
In how many ways can $15$ people be seated around two round tables with seating capacities of $7$ and $8$ people? |

✖ A. |
$\dfrac{15!}{8!}$ |

✖ B. |
$7!×8!$ |

✔ C. |
${^{15}C_8}×6!×7!$ |

✖ D. |
$2×{^{15}C_7}×6!×7!$ |

✖ E. |
${^{15}C_8} × 8!$ |

**Solution:**

Option(**C**) is correct

**Circular Permutation**

$n$ objects can be arranged around a circle in $(n - 1)!$

If arranging these $n$ objects clockwise or counter clockwise means one and the same, then the number arrangements will be half that number.

i.e., number of arrangements $=\dfrac{(n-1)!}{2}$

You can choose the $7$ people to sit in the first table in ${^{15}C_7}$ ways.

After selecting $7$ people for the table that can seat $7$ people, they can be seated in:

$(7-1)! = 6!$

The remaining $8$ people can be made to sit around the second circular table in:

$(8-1)! = 7!$ Ways.

Hence, total number of ways,

${^{15}C_8}× 6! × 7!$

**Anish**

*()
*

because 15c7 and 15c8 is same nCr=nC(n-r)

**Debasish Dey**

*()
*

Why n objects can be arranged around a circle in (n−1)! shouldn't it be n! ways?

In the procedure it is written 7 people can be chosen in 15c7 ways. but in the answer choices the option is given as 15c8. the same thing has been done in the soln. why?