TITLE: Six throws, only two distinct numbers: coincidence?
QUESTION [17 upvotes]: My daughter threw a die (D6) six times and got: 6,2,2,6,2,6.
This got us wondering whether having only two distinct numbers come up in six throws was very unlikely or would happen fairly often. How would we go about working out the answer to that question? (short of doing it lots of times and counting!)

REPLY [10 votes]: Here is an alternative solution to the (very good) answer already given here:
The number of ways that you can choose $2$ out of $6$ values is $\dbinom{6}{2}=15$.
Given $2$ values A and B, the number of combinations that you can get in $6$ rolls is:

Value A appearing $1$ time  and value B appearing $5$ times: $\dbinom{6}{1}=6$
Value A appearing $2$ times and value B appearing $4$ times: $\dbinom{6}{2}=15$
Value A appearing $3$ times and value B appearing $3$ times: $\dbinom{6}{3}=20$
Value A appearing $4$ times and value B appearing $2$ times: $\dbinom{6}{4}=15$
Value A appearing $5$ times and value B appearing $1$ time : $\dbinom{6}{5}=6$

So you can get $6+15+20+15+6=62$ combinations containing A and B.
And you can get $15\cdot62=930$ combinations containing any $2$ out of $6$ values.
The total number of combinations that you can get in $6$ rolls is simply $6^6=46656$.
Therefore, the probability of having exactly $2$ values in $6$ rolls is $\dfrac{930}{46656}\approx0.0199$.