You conduct a random experiment in which you toss a coin 10 times. How many possible outcomes with exactly 6 heads are there in this random experiment?

The number of possible outcomes with exactly 6 heads in 10 coin tosses can be calculated using the binomial coefficient, which is a key part of the binomial theorem. The binomial coefficient is often read as "n choose k", where n is the total number of events (in this case, 10 coin tosses) and k is the number of successful outcomes we're interested in (in this case, 6 heads).

The formula for the binomial coefficient is:

C(n, k) = n! / [k!(n-k)!]

where "!" denotes factorial, meaning the product of all positive integers up to that number.

So, to calculate the number of possible outcomes with exactly 6 heads in 10 coin tosses, we plug our values into the formula:

C(10, 6) = 10! / [6!(10-6)!]

Calculating the factorials, we get:

C(10, 6) = 3,628,800 / [720 * 24]

Solving this, we find:

C(10, 6) = 210

So, there are 210 possible outcomes with exactly 6 heads in 10 coin tosses.