probability

Question

A coin is tossed eight times. How many ways can it come up heads 7 times and tails once?

Answer 1

The number of ways to choose one flip out of 8 to come up tails is ₈C₁ = ₈P_1,7 = 8!/1!7! = 8, where C stands for combinations and P stands for permutations.

 

Viewing this as a combinations problem, we are choosing from 8 flips, f1, f2, f3,..., f8 which one will come up tails.  Viewing it as a permutions problem, it is counting the number of ways to permute the sequence THHHHHHH where 7 things are similar and 1 is dissimilar.  In either case, the answer is the same.  8.

Answer 2

eight ways; tails can come up first, second, third, fourth, fifth, sixth, seventh, or eighth

wherever tails comes up, heads will be in all the other places

 

HHHHHHHT

HHHHHHTH

HHHHHTHH

HHHHTHHH

HHHTHHHH

HHTHHHHH

HTHHHHHH

THHHHHHH