Probabilities (Math)

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?

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  1. a) each turn could be 2 ways, so 2^14 = 16384

    b) this is the same as asking, "in how many ways can we arrange 9 H's and 5 T's, the H's and T's are indistinguishable.

    number of ways = 14!/(9!5!) = 2002

    c)at least two heads implies we don't want 0 heads, or 1 head, let's find those two
    0 heads ---> 1 way
    1 head -----> 14!/13! = 14
    so at least 2 heads = 16384 - 14 - 1 = 16369

    d) so we don't want 11 heads, 12 heads, 13 heads and 14 heads
    which are 14!/(11!3!) + 14!/(12!2!) + 14!/(13!1!) + 1
    = 364 + 91 + 14 + 1 = 470

    so at most 10 heads = 16384 - 470 = 15914

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