There are several questions similar to this one but after reading those, I am still very confused.
I also did a similar problem of this one and I think I got it, but then I got stuck again.
So if four dice are rolled, the chance of getting three of a kind is: $ \binom{6}{1} \frac{1}{6}* \binom{5}{1} \frac{1}{6}*\binom{4}{1} \frac{1}{6} *\frac{1}{6}$
so if seven dice are rolled, in my understanding, the chance of getting three of a kind and four of another would be: $ \binom{7}{1} \frac{1}{6}*\binom{6}{1} \frac{1}{6}*\binom{5}{1} \frac{1}{6} *\binom{4}{1} \frac{1}{6} *\binom{3}{1} \frac{1}{6} *\binom{2}{1} \frac{1}{6} *\binom{1}{1} \frac{1}{6} $ however, the answer in the book is $ \frac{6 *5*\binom{7}{4} }{6^7} $ and I am totally lost. Please help!
additional problems The answers you guys gave kind of make sense to me but they also make me very confused. can I think of it using the way I did above? for example, another part of the questions asked about the chance of getting two fours, two fives and three sixes. I think of it as: $ \binom{7}{2} \frac{1}{6}^2*\binom{5}{2} \frac{1}{6}^2*\binom{3}{3} \frac{1}{6}^3 $ which matches the solution in the book.