a) Since it doesn't tell you the number of people and only gives you "x", try using a few different numbers to see a pattern.
Eg let x=7
If 7 ppl are in a line and 3 of them are in a group (together) it would be something lik: P P P G G G P
where P is a random person and G are the members of the group. Since the 3Gs make up one group, the total number of groups in the above line is 5 because there's 4 separate Ps and 1 group of 3, so in total it's 4+1 groups which is 5.
If you have 5 groups, firstly find the number of ways they can exist, which is just
5!
Then you arrange the G members in their own separate group who are all toghether, which ca be done
3! ways since there's 3 of the G people.
So in total there are
5! x 3! arrangements for
7 people
Another example: if you have 6 people eg P G G G P P
The Gs need to be together, so that's one group then there's 3 separate groups of P. In total there are 4 groups which can be arranged in
4! ways.
Then the group of 3 G people can be arranged in
3! ways.
So the total number of arrangements is
4! x 3! for
6 people.
Also, if you have 5 people in total there are 3! x 3! arrangements
Using that pattern we can see that for 7 people the tot number of arrangements was 5! x 3! = 3!(7-2)!
and for 6 people it was 4! x 3! = 3!(6-2)!
For 5 people it was 3! x 3! = 3!(5-2)!
so for "x" number of people it will be 3!(x-2)! using the pattern from above.
See if you can do the next part
