An experiment consists of tossing a coin 14 times.

(a) How many different outcomes are possible?
(b) How many different outcomes have exactly 9 heads?
(c) How many different outcomes have at least 2 heads?
(d) How many different outcomes have at most 10 heads?

To answer these questions, we need to understand the concept of combinations and permutations.

In this coin-tossing experiment, we have 2 possible outcomes for each toss: either a head (H) or a tail (T). So, there are 2 outcomes for each toss.

(a) To calculate the total number of different outcomes for 14 coin tosses, we need to consider that each coin toss is an independent event. The total number of different outcomes is obtained by multiplying the number of outcomes for each toss together. Since we have 2 possible outcomes (H or T) for each of the 14 tosses, we can use the multiplication rule. Therefore, the total number of different outcomes is 2^14 = 16,384.

(b) To determine the number of different outcomes that have exactly 9 heads, we need to find the number of combinations of 9 heads out of 14 tosses. The formula for combinations is nCr = n! / (r!(n-r)!), where n is the total number of trials, and r is the number of successful outcomes. In this case, n = 14 and r = 9. Plugging in these values, we get 14C9 = 14! / (9!(14-9)!) = 2002. Therefore, there are 2002 different outcomes that have exactly 9 heads.

(c) To find the number of different outcomes that have at least 2 heads, we can calculate the number of outcomes with 0 and 1 head, and subtracting it from the total number of outcomes.

For 0 heads, there is only 1 possible outcome (TTTTTTTTTTTTTT).
For 1 head, there are 14 possible outcomes (HTTTTTTTTTTTTT, THTTTTTTTTTTTT, TTHTTTTTTTTTTT, etc.).

So, there are 1 + 14 = 15 outcomes with 0 or 1 head. Subtracting this from the total number of outcomes (16,384) gives us 16,384 - 15 = 16,369 different outcomes that have at least 2 heads.

(d) To calculate the number of different outcomes that have at most 10 heads, we need to add up the number of outcomes with 0, 1, 2, ..., 10 heads.

For 0 heads, like we mentioned earlier, there is only 1 possible outcome.
For 1 head, there are 14 possible outcomes.
For 2 heads, there are 14C2 = 91 possible outcomes.

Using the same formula, we can calculate the number of outcomes for 3, 4, 5, ..., 10 heads. Adding them together gives us the total number of outcomes with at most 10 heads.