A coin is tossed 8 times a How many possible outcomes sequen

A coin is tossed 8 times.

(a) How many possible outcomes (sequences of heads and tails) are there with 3 heads?

(b) How many different outcomes are there exactly 3 heads where the heads are non-consecutive?

Solution

a)

By permutation of like objects, there are 8!/(3!5!) = 56 such outcomes. [ANSWER, 56]

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b)

n(3 nonconsecutive heads) = n(3 heads) - n(3 consecutive heads)

Treating the 3 heads as one body, then there are just 6!/(1!5!) = 6 ways to arrange so that they are consecutive.

Thus,

n(3 nonconsecutive heads) = n(3 heads) - n(3 consecutive heads)

n(3 nonconsecutive heads) = 56 - 6

n(3 nonconsecutive heads) = 50 [ANSWER]