finding numbers of arrangments (please check my work)

[Clifford]

I was wondering if somebody could take a look over these?

1. How many ways can a photographer arrange 7 scouts, 6 cubs and 5 leaders if there are no restrictions?

2. How many ways can a photographer arrange 7 scouts, 6 cubs and 5 leaders, if the leaders sit in the first row, the cubs sit in the second row and the scouts stand in the third row?

3. How many arrangement are there if the tallest leader, the two tallest cubs and the three tallest scouts are to be at the centre of their row?

Solution:

1. (7+6+5)! = 18! = 6402373705728000

2. There are 7! ways to arrange scouts, 6! for cubs, and 5! for leaders
7! * 6! * 5! = 435456000

3. If one leader is in the centre, take one off the total and arrange
(5-1)! = 4! = 24
take 2 cubs off the total and arrange
(6-2)! = 4! = 24
take 3 scouts off total and arrange
(7-3)! = 4! = 24
multiply together
24 * 24 * 24 = 13824

 

[galactus]

Re: arrangments

deleted.

I was afraid of that. I overlooked the 'no restrictions' in the problem statement.

 

[Clifford]

It says there are no restrctions though, that is why I just did 18! even though there are xx of a certain group, they are different people.

Are the rest ok?

 

[pka]

I agree with you that there are eighteen individuals so #1 is (18!).

The #2 is correct.

Assuming that the seating in #3 follows the pattern in #2, then:
Row 1 is (4!).
Row 2 is (2!)(4!). The two tallest can stand in two ways.
Row 3 is (3!)(4!). The three tallest can arrange themselves in (3!) ways.