Question

# In how many ways may 6 Hindus and 6 Muslim’s sit around a round table so that 2 Hindus may never sit together?

Hint:Fix one person, either a Hindu / Muslim in order to calculate where other’s sit. Then use gap method in circular permutation and combination and find the number of seating arrangements for Hindu’s and Muslim’s. Multiply them together to get the number of way to arrange them.

It’s given in the question that there are 6 Hindus and 6 Muslims.
We have to arrange 12 people around a round table.
Let us fix one Muslim person in a seat. Now there are 5 more Muslim persons to be seated. Now, by using the gap method in circular permutation or combinations.
We can arrange the remaining 5 Muslims in $5!$ ways.
$5!=5\times 4\times 3\times 2\times 1=120$ ways -(1)
Now, it is given that we can’t arrange seating in a way that two Hindus come together.
6 Hindu’s can be arranged in 6 empty seats between two consecutive Muslims. By this way, no two Hindus will come together.
We can arrange the six Hindus in 6! Ways.
$6!=6\times 5\times 4\times 3\times 2\times 1=720$ ways – (2)
So, now we need to find the total number of ways in which we can arrange 6 Hindus and 6 Muslims to sit around a round table.
Equation (1) and equation (2) should be multiplied to get the desired ways.
$\therefore$Total number of ways $=5!\times 6!=120\times 720=86400$
$\therefore$Number of ways to arrange 6 Hindus and 6 Muslims is equal to 86400 ways.

Note: One person either Hindu/Muslim should be fixed first in order to calculate where other’s sit. Here, order is not necessary.Student should fix a particular person first and later we have to find an arrangement based on given conditions.