How many ways can the company assign three deliveries to its couriers?

Question

(b) On a given afternoon, a delivery company must make three deliveries to separate customers. Five couriers are available.

Two of the couriers are trainees.

(i) In how many ways can the couriers be allocated to the deliveries? (4 marks)

(ii) In how many ways can the couriers be allocated to the deliveries using only experienced couriers? (4 marks)

There are 5 delivery drivers and three parcels.

Two are trainees

i) $5C3 = 10$

As order does not matter it can be a combination

$5$ drivers C $3$ parcels

ii) $3P3 = 6$

The reason I have choose permutation for this question is that the order matters because the two trainees that can not be chosen.

I am really not sure how to approach these problems.

Answer 1

On a given afternoon, a delivery company must make three deliveries to separate customers. Five couriers are available. In how many ways can the couriers be allocated to the deliveries?

It matters which courier makes which delivery. If the company assigns a different courier to each delivery, then there are $P(5, 3) = 5 \cdot 4 \cdot 3 = 60$ possible assignments. If each delivery may be assigned to any of the couriers, then there are five choices for each delivery, so there are $5^3 = 125$ possible assignments.

On a given afternoon, a delivery company must make three deliveries to separate customers. Five couriers are available. Two of the couriers are trainees. In how many ways can the couriers be allocated to the deliveries using only experienced couriers?

Again, it matters which courier makes which delivery. If the company assigns a different experienced courier to each delivery, then there are $P(3, 3) = 3 \cdot 2 \cdot 1 = 6$ possible assignments, as you found. If each delivery can be assigned to any of the experienced couriers, there are three choices for each delivery, so there are $3^3 = 27$ possible assignments.