In how many ways can...

Question

In how many ways can 5 different contracts be distributed among 12 different firms; if no two contracts are awarded to the same firm?

Answer 1

The first step is to select the 5 firms that will be awarded a contract (since a firm can't receive more than 1 contract). How many ways can you select 5 elements from a set of 12 elements?

 

The next step is to calculate the number of ways you can assign the 5 contracts to the 5 firms that were selected. First assign an order to the 5 firms that were selected (i.e., firm 1, firm 2, firm 3, firm 4, firm 5). Now you have a list of 5 firms where the order matters. How many ways can you assign the 5 contracts to these firms?

 

Once you have those two values, multiply them together to get the total number of ways you can assign 5 contracts to a pool of 12 firms where no single firm is allowed more than 1 contract. This question boils down to combinations AND permutations. The first step is calculating combinations, while the second step is calculating the number of permutations for a given combination of firms.