Question

A party has 20 participants. Find the number
of distinct ways for the host to sit with them
around a circular table. How many of these
ways have two specified persons on either
side of the host?​

Answer 1

Answer:

!20/!2×!18

Step-by-step explanation:

20c_2=!20/!2×!(20-2) = !20/!2×!18

Answer 2

There are 18 ! x 2 ways to have two specified persons on either  side of the host

Solution:

A party has 20 participants

Along with 1 host, there are 21 people

Two particular persons are seated on either side of the host

Let us consider the host and these two particular persons as one entity

Now we have, (21 - 3 + 1) persons

21 - 3 + 1 = 21 - 2 = 19

19 people are to be seated around the circular table

Number of ways of arranging n people around a circular table is (n - 1)!

This can be done in (19 - 1)! ways = 18 !

Two particular persons who are seated on either side of host can interchange their positions

This can done in 2 ! ways = 2 ways

Hence total number of ways = $18 ! \times 2$

Thus, there are 18 ! x 2 ways to have two specified persons on either  side of the host

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