A bats man scores exactly a century lb hitting fours and sixes in 20 consecutive balls. In how many different ways can e do it if some balls may not yield runs and the order of boundaries and over boundaries are taken into account?

Explanation

Let the batsman hit x fours, y sixes and let z balls may not yield runs. Then we have
4x+6y+0z=100
where
x+y+z=20
From Eqs. (1) and (2),
4(20-y-z)+6y=100
or 2y-4z=20
or y-2z=10
or y=10+2z
hence, z can be 0,1,2,3 as any of 4x and 6y cannot exceed 100. So, we have following possible distribution of runs :

Hence, the total number of ways is
`20!((1)/(16!3!)+(1)/(14!4!2!)+(1)/(12!7!)+(1)/(10!10!))`