To answer these questions, we will use the concept of permutations and combinations.
(a) To find the number of different outcomes possible when tossing a coin 14 times, we need to consider each toss as an independent event. Since there are two possible outcomes (heads or tails) for each toss, we can use the formula for the total number of outcomes in multiple events:
Total outcomes = (Number of outcomes for 1st event) x (Number of outcomes for 2nd event) x ... x (Number of outcomes for 14th event)
In this case, the number of outcomes for each event is 2 (heads or tails). Therefore, the total number of different outcomes possible when tossing a coin 14 times is 2^14 = 16,384.
(b) To find the number of different outcomes that have exactly 9 heads, we need to choose 9 out of the 14 tosses to be heads and the remaining 5 tosses to be tails. We can use the combination formula:
Number of outcomes with exactly 9 heads = C(14, 9) = 14! / (9! * (14-9)!) = 2002
(c) To find the number of different outcomes that have at least 2 heads, we need to consider two cases: one with exactly 2 heads and the other with more than 2 heads.
For the case with exactly 2 heads, we need to choose 2 out of the 14 tosses to be heads and the remaining 12 tosses to be tails:
Number of outcomes with exactly 2 heads = C(14, 2) = 14! / (2! * (14-2)!) = 91
For the case with more than 2 heads, we can use the concept of complement:
Number of outcomes with at least 2 heads = Total outcomes - Number of outcomes with no heads or 1 head
Number of outcomes with no heads = C(14, 0) = 1
Number of outcomes with 1 head = C(14, 1) = 14
Number of outcomes with at least 2 heads = 16,384 - 1 - 14 = 16,369
(d) To find the number of different outcomes that have at most 10 heads, we need to consider two cases: one with 0 to 10 heads.
Number of outcomes with 0 heads = C(14, 0) = 1
Number of outcomes with 1 head = C(14, 1) = 14
Number of outcomes with 2 heads = C(14, 2) = 91
Number of outcomes with 3 heads = C(14, 3) = 364
Number of outcomes with 4 heads = C(14, 4) = 1001
Number of outcomes with 5 heads = C(14, 5) = 2002
Number of outcomes with 6 heads = C(14, 6) = 3003
Number of outcomes with 7 heads = C(14, 7) = 3432
Number of outcomes with 8 heads = C(14, 8) = 3003
Number of outcomes with 9 heads = C(14, 9) = 2002
Number of outcomes with 10 heads = C(14, 10) = 1001
Number of outcomes with at most 10 heads = Sum of outcomes with 0 to 10 heads = 1 + 14 + 91 + 364 + 1001 + 2002 + 3003 + 3432 + 3003 + 2002 + 1001 = 13,568