If three corporations are competing for four different contracts, what is the probability that each corporation will get at least one contract?

Question

Using Bayes' Theorem, how would I go about solving this?

Suppose that there are three corporations competing for four different contracts. If the contracts are awarded randomly, what is the probability that each corporation will get at least one contract?

Would I do:

$$\frac{(4!)\cdot(3)}{2}$$

My book had used a problem similar to this but I'm not sure. Thanks!

Answer 1

The number of ways for each corporation to get at least one contract is $$4!\cdot 3$$ and the number of ways for the contracts to be awarded randomly is $$3^4$$ Thus, the probability is $$\frac{4!\cdot 3}{3^4}$$

Answer 2

The favoured event consists of one corporation with two contrats and two with one each.

What is the probability for selecting two from four contracts, a corportation for them, and arrange the remaining corportations between the remaining two contracts, when selecting from three corportations (with repeatition) for each of four contracts?

$$\dfrac{\binom 42\binom 31 2!}{3^4}~=~\frac 49$$