Question

a nursery school teacher has 4 apples 3 oranges and 3 bananas to share among 9 children, with each child receiving one fruit. Find the number of different ways in which this can be done.

Answer 1

There are 10 fruit but only 9 children.
If each child is to get one fruit there are only a few cases:
4a , 3o, and 2b
4a, 2o and 3b
3a , 3o and 3b , I see no other cases.

In each of the 3 cases, we have 9 fruits, but in each case we could have arrangements of the fruits amongst the 9 children.

for 4a,3o,2b we can arrange that in 9!/(4!3!2!) ways or 1260 ways
for 4a,2o,3b, we have the same, or 1260 ways
for 3a,3o,3b we have 9!/(3!3!3!) or 1680 ways

Number of different ways is 1260+1260+1680 = 4200

check my arithmetic.