What is the probability that the number of heads is equal to the number showing on the die? For this question, I know how to calculate the sample space, but I'm not sure how to use permutations or combinations. Here is what I have so far. Can anyone please help me out?
Suppose we roll one fair six-sided die, and flip six coins. What is the probability that the number of heads is equal to the number showing on the die?
$S = 6 + 2^6 = 70$
$P(1H, 1) = \frac{7!}{5!2!*70}$
$P(2H, 2) = \frac{7!}{4!3!70}$
$P(3H, 3) = \frac{7!}{4!3!*70}$
$P(4H, 4) = \frac{7!}{5!2!*70}$
$P(5H, 5) = \frac{7!}{6!*70}$
$P(6H, 6) = \frac{7!}{70}$
$ Total = P(1H, 1) + P(2H, 2) + P(3H, 3) +P(4H, 4) +P(5H, 5) + P(6H, 6)$
 A: The sample space is the product of the sample spaces for the dice and the coins.  If you order the coins the space is $6 \cdot 2^6=384$  
Now consider the number of ways to get each total of heads from $1$ to $6$.  They should add to $63$ because you don't consider the $1$ way to get no heads.  For each of them there is one favorable roll of the die, so there are $63$ favorable combinations.  The probability is then $$\frac {63}{384}=\frac {21}{128}$$
A: You seem to be using Binomial coefficients $\large 7\choose k$ for different values of $k$, which would be relevant if you were repeating some binary experiment $7$ times identically and independently, which you are not. One of your seven experiments is drastically different from the others. It is independent from the other six though, which means that you may compute probabilities separately and multiply them in the end ($P(A\cap B)=P(A)\cdot P(B)$ for independent events $A,B$).
Bottom line: use the binomial approach for the coin and then multiply with the probability of getting a given number when you roll a dice.
