Finding patterns 9 nine items of two types. There was a question in a mock test of an olympiad I gave, which says,
"In how many ways 5 humans and 4 monkeys be seated around a table, such that no two monkeys sit together"
Now, in this question how can I deal with the statement that no two monkeys can sit together. 
 A: Hint first arrange the humans around the table alternatively. This can be done in $5!$ ways . Now automatically no two monkeys are together so they can be arranged in $4!$ ways hence total ways are $5!.4!$ 
A: Take the monkeys to be distinguishable, the table seats to be distinguishable, and the humans to be  distinguishable.
No two monkeys can sit together implies that every monkey has a human on his right (if this was a linear table rather than circular, an end case would have to be considered separately).  So we can glue a human to each monkey, thiws can be done in $5\cdot4\cdot3\cdot2=120$ ways. Lable these glued pairs A, B, C, D
Now we can consider pair A as "first" and order pairs B, C, and D in $6$ ways.  Having done that, we can decide to slot the leftover human in 4 ways (before A is the same as after D).
Finally, we can start pair A in $9$ possible monkey seats.
$$
120 \cdot 6 \cdot 4 \cdot 9 = 25920
$$
arrangements.
A: I tried to visualize the question and I find the only combination of no two monkeys can sit together is as this(https://i.imgsafe.org/a16562328c.jpg)  
Because if you want to switch the seats of monkeys, like this(https://i.imgsafe.org/a170d53318.jpg)
It would still be the exact same permutation as the first one( you can visualize it by rotating the picture).  
So only one permutation is allowed, and assume that we have different people and different monkeys, the number of ways would be $5!*4!$
A: Unless otherwise specified, seats at a round table are treated as unnumbered,
hence the formula $(n-1)!$
Taking each living being to be distinct, seat the "chief" somewhere among five chairs,
and arrange the other humans in $4!$ ways.
Four chairs for the monkeys can be inserted in the $5$ gaps between humans $\binom54$ ways, and the monkeys now seated in $4!$ ways  
Putting the pieces together, $4!\binom544! = 2880$ ways 
NOTE:
Although your header mentions "patterns", there is no such word in the question.
