UBS Television Company is considering bids submitted by twelve different firms for three different contracts. In how many ways can the contracts be awarded among these firms if no firm is to receive more than two contracts?

Question

UBS Television Company is considering bids submitted by twelve different firms for three different contracts. In how many ways can the contracts be awarded among these firms if no firm is to receive more than two contracts? ​

Quizplus · Accepted Answer

Since no firm can receive more than two contracts, the three contracts can be divided in four ways: 
1. One firm receives two contracts, two firms receive one contract each. There are 12 options for selecting the firm receiving two contracts, and 11 options for selecting the firms receiving one contract each (after the first firm has been selected). The total number of possibilities is 12*11*10. 
2. Two firms receive two contracts each, one firm receives one contract. There are 12 options for selecting the firm receiving one contract, and 11 options for selecting the first firm receiving two contracts (after the first firm has been selected). For each of these options, there are 10 options for selecting the second firm receiving two contracts (after the first firm has been selected). The total number of possibilities is 12*11*10. 
3. Three firms receive one contract each. There are 12 options for selecting the firm receiving the first contract, 11 options for selecting the firm receiving the second contract (after the first firm has been selected), and 10 options for selecting the firm receiving the third contract. The total number of possibilities is 12*11*10. 
4. One firm receives one contract, two firms receive two contracts each. There are 12 options for selecting the firm receiving one contract, and 11 options for selecting the first firm receiving two contracts (after the first firm has been selected). For each of these options, there are 10 options for selecting the second firm receiving two contracts (after the first firm has been selected). The total number of possibilities is 12*11*10.

Therefore, the total number of ways the contracts can be awarded among these firms is: 
12*11*10 + 12*11*10 + 12*11*10 + 12*11*10 = 5280

Therefore, C is the correct answer.

UBS Television Company Is Considering Bids Submitted by Twelve Different