Perms and Combs question (arrangements in a circle) (1 Viewer)

[Wonder]

From Cambridge Yr 12 3 unit, Exercise 10I
A committee of 3 women and 7 men is to be seated randomly at a round table.
a) What is the probability that the 3 females will sit together?
b) The committee elects a president and vice-president. What is the probability that they are sitting opposite one another?

And I'm having difficulty understanding probability in arrangements around a circle in general...if someone could try to explain how it works in general, with this example?

Thank you!

 

[braintic]

From Cambridge Yr 12 3 unit, Exercise 10I
A committee of 3 women and 7 men is to be seated randomly at a round table.
a) What is the probability that the 3 females will sit together?
b) The committee elects a president and vice-president. What is the probability that they are sitting opposite one another?

And I'm having difficulty understanding probability in arrangements around a circle in general...if someone could try to explain how it works in general, with this example?

Thank you!

(a) Total number of arrangements = 9!
Treating the women as a block, number of arrangements = 7! times 3!
so 7! times 3! / 9!

(b) Once the president is seated, there are 9 seats left for the VP, of which one is directly opposite.
So 1/9

 

[Wonder]

(a) Total number of arrangements = 9!
Treating the women as a block, number of arrangements = 7! times 3!
so 7! times 3! / 9!

(b) Once the president is seated, there are 9 seats left for the VP, of which one is directly opposite.
So 1/9

This is part that I'm having trouble understanding...if you could expand on it?
Thanks!

 

[braintic]

This is part that I'm having trouble understanding...if you could expand on it?
Thanks!

The women have to sit together, so treat them as one block. So there are 8 'items' to be moved around the circle: The 7 men plus the block of women.
The number of ways of arranging 8 things in a circle is 7!.
But then the women can be arranged with the block, so times by 3!.

Another way to think about it: Pick the leftmost seat that the women will be sitting in. If you only care about relative positions, the position of this seat is irrelevant. Once you've picked the leftmost seat, the other 2 seats for women are picked for you.
Now arrange the women within these seats: 3! ways.
Now place the 7 men in the remaining spots: 7! ways.
Same answer.
Total no. of ways = 3! times 7!

 

[Wonder]

Got it! Thank you

 

[braintic]

Got it! Thank you

Actually my last line was wrong. Its now correct.