# Discrete Mechatronics - sequences with repeats and no repeats

We have been given this task - Some of it I did with quite and ease while others I add issues with. The ones I answered were a combined group efforts of me and my friends so I'd like you guys' explanation on them since you probably know better.

Here it is:

How many ways are there to seat down 40 students on top of 10 different benches of length 4?

My answer - This one is ok. this could have been a 40 length bench for all we care so $40!$

b. How many ways are there to seat down 40 students on top of 10 different benches of length 4, such that Sia sits at the end of some bench (doesn’t matter which), and beside her sits Kendrik or Rihanna?

My answer - Sia has 20 possible places to sit. Afterwards either Kendrik or Rihanna can sit next to her. This gives us $20*2*38!$

c. How many ways are there to seat down 40 students on top of 8 different benches of length 5, such that Ariana and Nicki do not sit on the same bench, and Alessia sits at the center of the last bench?

My answer - Alessia has one possible place to sit. We basically remove her from the equation. What we eventually get to is $34*35*37!$. Not quite sure how to arrive to this and whether it is actually true

d. How many ways are there to seat down 40 students on top of 10 different benches of length 4, such that Beyonce sits between Kanye and Taylor?

My answer - This is worded quite badly. I believe they mean that Beyonce has to sit strictly between Kanye and Taylor so that a bench would look like "xKBT"/xTBK" or "KBTx"/"TBKx". but not "KxBT"...

We arrived at $40*37!$ although again I'm not sure about the process or the answer

e. How many ways are there to seat down 6 girls and 6 boys on 6 different benches for two, such that on every bench there is one boy and one girl?

My answer - I arrived at $(6!)^2$. Is this the way to go?

f. How many ways are there to seat down five boys and five girls, on two parallel benches of length 5, such that there is at least one girl opposite in front of a boy?

My answer - None. Can you help?

g. How many ways are there to seat down 8 girls and 20 boys in two circles: in the circle further from the door are all the girls, and in the circle closer to the door are all the boys and Drake sits closest to the door?

My answer - None. Can you help?

Thank you!

• Better post only the ones that you're unsure of or have difficulties with, and in separate posts (one question per post). Dec 22, 2016 at 8:43
• I didn't want to spam all of them (Pretty knew to this). Thanks for the tip I'd definitely do this next time
– RonH
Dec 22, 2016 at 8:55
• at e), would sitting left and right matter? Dec 22, 2016 at 9:27
• at f) with odd number, wouldn't there always be at least one girl and one boy opposite to eachother? Dec 22, 2016 at 9:27

Your first 3 answers and explanations are perfect.

a) Answer is 40!.

b) Answer is 20 * 2 * 38!.

c) Answer is 34 * 35 * 37!.

d) Beyonce cannot sit on any end so we left with 20 seats. So he can sit on any seat. Kanye and Taylor sit on left right of Beyonce in 2 ways.

And remaining students can sit in 37! ways.

Answer is 2 * 20 * 37!.

e) Either boys can sit on right or left.

Answer is 2^6 * (6!)^2.

f) There is no case in which no girl is opposite to boy. As number of boys and girls are odd.