(a) \({}_{12}{C_5} = 792\); There are 792 ways that five different #s can be chosen from the numbers 1 to 12.(b) (i) The number of ways that three winning numbers can be chosen from the five winning numbers is \({}_5{C_3} = 10\); and the number of ways that two non-winning numbers can be chosen from the seven non-winning numbers is \({}_7{C_2} = 21\). Thus, the number of ways that three of the five winning numbers can be chosen is \({}_5{C_3} \cdot {}_7{C_2} = 10 \cdot 21 = 210\). Therefore,...