A popular gambling game called keno first introduced in Chin

A popular gambling game called keno, first introduced in China over 2000 years ago, is played in many casinos. In keno, there are 80 balls numbered from 1 to 80. The casino randomly chooses 20 balls from the 80 balls. These are \"lucky balls\" because if a gambler chooses some of the numbers on these balls, there is a possibility of winning money. The amount that is won depends on the number of lucky numbers the gambler has selected. The number of ways in which a casino can choose 20 balls from 80 is Once the casino chooses the 20 lucky balls, the remaining 60 balls are unlucky for the gambler. A gambler who chooses 5 numbers will have from 0 to 5 lucky numbers. Let\'s consider the case in which 2 of the 5 numbers chosen by the gambler are lucky numbers. Because 5 numbers were chosen, there must be 3 unlucky numbers among the 5 numbers. The number of ways of choosing 2 lucky numbers from 20 lucky numbers is .The number of ways of choosing 3 unlucky numbers from 60 unlucky numbers is. By the counting principle, there are to choose 2 lucky and 3 unlucky numbers.

Solution

The number of ways selecting 1 lucky number out of 20 is C(20,1)

The number of ways of selecting remaining 5 unlucky numbers out of 60 is C(60,5)

And C(n,r)=n!/(r!(n-r)!)

C(20,1)=20!/(1!(20-1)!)=20!/(1!*19!)=20

And C(60,5)=60!/(5!(60-5)!)

                     =60!/(5!* 55!)= 5461512

Therefore numberof ways of selecting 1 lucky number =C(20,1)*C(60,5)

                                                                                           =20*5461512=109230240