Discrete mathematics 5 how many ways can be distributed eigh

Discrete mathematics

5.- how many ways can be distributed eight chocolate donuts and seven glazed donuts on three students if everyone wants at least one of each type.

Solution

total we have 15 chocolates and 3 students

where 15 chocolates = 8+ 7

where each student have atleast 1 of the chocoltes

so we have ( 2,2,2) minimum for each student where 1 from each variety

and the remaining 9 can be given in different ways such as ( 1,1,7)(1,2,6) (1,3,5) (1,4,4)(2,2,5) (2,3,4) (3,3,3)

where each combination is again re arranged in different ways

(1,1,7) in 3 ways such as ( 1,1,7) (1,7,1) (7,1,1)

similrly (1,4,4,) (2,2,5) in 3 ways totally 3 x 3 = 9ways

(1,2,3) & (1,3,5)& (2,3,4) in 6 ways totally 3X6 = 18 ways   and ( 3,3,3) in 1 way

total no of ways = 18 + 9 + 1 = 28 ways

no of ways we can distribute 8 and 7 chocolates of 2 vrieties to 3 students where each student get atlest one of each type is 28