A standard deck of cards has had all the face cards (jacks, queens, and kings) removed so that only the ace (given a value of 1) through ten of each suit remains. A game is played in which two cards are drawn (without replacement) from this deck and a six-sided die is rolled. Find the probability that the sum of the cards and the die is 7.

This is the solution in the book:

I'm not sure if there's an easier way, but that has been a little confused. For starters, why is the (3,3,1) separated from the top set? Also, shouldn't the first part be (16/40)(16/39)(1/6)?

The way I attempted to set it up was like this:

(1,5,1)(2,4,1)(3,3,1)(4,2,1)(5,1,1)

+ (1,4,2)(2,3,2)(3,2,2)(4,1,2)

+ (1,3,3)(2,2,3)(3,1,3)

+ (1,2,4)(2,1,4)

+ (1,1,5)

Can anyone clarify the book solution? Why is it being done like that opposed to what I had done? Thanks a lot for the help!