Permutations with 2 items in adjacent rows and further restriction 2 items not in adjacent rows

[ry7]

Can someone tell me if I am reasoning through this correctly?
The question series is (A) A farmer is planting 7 crops, with the restriction that 2 of the crops, carrots and asparagus, must be in adjacent rows.
(B) Redo (A) with the extra restriction that peas and Brussels sprouts NOT be in adjacent rows
(C) Redo (B) with the further restriction that the 7 crops be planted in a circle.

(A)




(B)




(C)

 

[ry7]

I changed my thinking on the final question, so I calculated the circular arrangement for carrots/ tomatoes adjacent, then subtracted carrots/ tomatoes adjacent and peas and beans adjacent from that total

 

[Steven G]

The first two look fine.

 

[Dr.Peterson]

Part C is a little odd, as it's unclear exactly what adjacent rows are when you plant in a circle. But I don't see how to take it other than what you show.

I also find it curious that asparagus and Brussels sprouts in the problem seem to have transmuted to tomatoes and beans in your answers.

Ignoring all that ...

Your second method looks good to me. I'd wait for a second person to confirm that, as these can be tricky. (The second person could be you, if you find another way to do it.)

 

[ry7]

Lol, you’re right I named the crops wrong, i’ve been turning this one around in my mind for a bit. What I’m trying to figure out is how to calculate a permutation when one of the restrictions is that 2 item NOT be adjacent- the only way I can think of to do that is to find the complement ie when they are, and subtract that. I’m clear on the calculating 2 things adjacent - treat them as a unit. And clear on the permutations in a circle (‘fix’ one unit in place, so(n-1)!
Thank you Dr. P! it makes sense to me this way... but I might leave it for a bit and revisit