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.
Best Answer
There are $6^7$ possible outcomes, listed as 7-tuples. Now, how many 7-tuples would consist of two distinct elements, 4 of one kind, and 3 of other. Such 7-tuples can be ordered into $[a,a,a,a,b,b,b]$ with $a \not= b$. $a$ can be chosen 6 different ways. Once $a$ is chosen, $b$ can be chosen 5 different ways. There are $\binom{7}{4}$ ways to rearrange $[a,a,a,a,b,b,b]$ tuple.
The probability is then the ratio of the number of needed outcomes, over the number of total outcomes: $$ p = \frac{1}{6^7} \binom{7}{4} 6 \times 5 $$