Combination Problem - Can anyone solve the question? (Hint, answer: 3072)

A child is told she can bring five toys with her on holiday. the child decides to choose the five from
6 jigsaws, 8 dolls, 4 balls, 2 trucks.
How many of these sets have at least one from each of the four categories of toy listed above?

The possibilities are JJDBT or JDDBT or JDBBT or JDBTT
Number of ways of selecting r number of items out n is given by nCr = n!/(r!*(n-r)!)
Total number of ways of selecting 5 toys = 6C2*8C1*4C1*2C1 + 6C1*8C2*4C1*2C1 + 6C1*8C1* 4C2*2C1 + 6C1*8C1*4C1*2C2
= 15*8*4*2 + 6*28*4*2 + 6*8*6*2 + 6*8*4*1
= 960 + 1344 + 576 + 192
= 3072

6 choices for jigsaws times
8 choices for dolls time
4 choices for balls times
2 choices for trucks.
that's 96•4 = 384 choices for 1 of each, which leaves 16 choices for the 5th toy, and
384 • 16 = 6144
which is twice your "answer"