Susan's friends Vicki, Louis, Maddy, Hank, and Rob have volunteered to help with the preparations for a party. How many ways can Susan assign someone to buy beverages, someone to arrange for food, and someone to send invitations? Assume that no person does two jobs.

There are only three tasks and 5 people. The number of ways of selecting 3 out of 5 people to perform these tasks is also the number of ways of (de-)selecting 2 out of 5, which is 10. Having selected the people we now have to work out who does what. There are 6 ways of arranging 3 tasks. The selected people can be assigned their tasks in 6 ways, so in total there are 6*10=60 ways of selecting and organising 3 out of 5 people to carry out three tasks.

To show this as a tree diagram, the first set of branches will be all the ways of selecting the people (shown by their initial letters):

on the V main branch we have 3 sub-branches: L, M, H;

on the V-L sub-branch we have 3 twigs: M, H, R;

on the V-M sub-branch we have 2 twigs: H, R;

on the V-H sub-branch we put twig R.

On the L main branch we have 2 sub-branches: M, H;

on the L-M sub-branch we have 2 twigs: H, R;

on the L-H sub-branch we have twig R.

On the M main branch we have one sub-branch H and one twig R on that branch.

This arrangements gives us 10 twigs, corresponding to the 10 combinations of people. I've taken the people in order of appearance: V, L, M, H, R, but that was arbitrary. We could have chosen any order, and we would still have ended up with 10 twigs. The main branch V involves L, M, H, R, those people further "along the line" so to speak. L involves M, H, R; so it's only necessary to involve people further down the line, because, for example, the combinations involving V and M have already been covered by the main branch V.

Each person in the group of three can be assigned one of three tasks; that leaves 2 tasks for the remaining two people. There are two ways two people can be assigned two tasks, so there are 3*2=6 ways for three people to be assigned three tasks. We can attach these 6 ways: b for beverage, f for food and i for invitations: bfi, bif, fbi, fib, ibf, ifb. These 6 are the leaves on each of the 10 twigs. By tracking along each branch, twig and leaf we have all 60 possible arrangements.

by Top Rated User (982k points)