If you flip 10 pennies at once, what are the chances
3 will come up heads and 7 will come up tails? Answer
is 15/128 – but how do you get that answer?

Suppose we number the pennies 1 thru 10. 
Let's look at a sample successful case.

penny #   1  2  3  4  5  6  7  8  9 10
          T  T  H  T  T  H  T  H  T  T

Let's count all the successful ways:

Of the ten numbered pennies we can choose the pennies
which are to come up heads any of the combinations of
10 things taken 3 at a time, that is   

10C3 or C(10,3) or 10!/(3!7!) or 120 way

Now let's count all the possible ways, successful or 
unsuccessful:

Each penny may come up either of 2 ways, so that's a 
total of 2×2×2×2×2×2×2×2×2×2 or 1024 ways

So the probability is 120/1024 = 15/128.

Another way:

This is a binomial probability of 3 successes out of
10 independent trials, with probability of 1/2 on each 
trial.

The formula for x successes out of n independent trials, 
with probability of success p is given by the formula:

n!/[x!(n-x)!]·px(1-p)n-x

This problem has n = 10, x = 3, p = 1/5

10!/[3!(10-3)!]·(1/2)x(1-p)n-x = .1171875 = 15/128

Another way

On your TI-83 or better calculator:

Press 

CLEAR, 2nd, VARS (to get DISTR menu)

scroll down to binompdf(

after binomcdf(,  type 10,1/2,3)

so that you see this on the screen

binompdf(10,1/2,3)

press  ENTER

read .1171875, which is the decimal value
of the answer

then press

MATH ENTER ENTER

read 15/128, the fraction answer.

Edwin