Question

A coin is flipped five times. Find the number of possible sets of heads and tails that have at most 3 heads

Answer 1

`Sets = 27`

Explanation:

Given

`Coin = 1`

`Toss = 5`

Required

Number of outcomes with at most 3 heads

First, we list out the sample space of a toss of coin 5 times

`(HHHHH)`,
`(HHHHT)`,
`(HHHTT)`,
`(HHTTT)`,
`(HTTTT)`,
`(TTTTT)`,
`(TTTTH)`,
`(TTTHH)`,
`(TTHHH)`,
`(THHHH)`,
`(HTHTH)`,
`(THTHT)`,
`(HHTHH)`,
`(TTHTT)`,
`(HTTHT)`,
`(THHTH)`,
`(THHHT)`,
`(HTTTH)`,
`(THHTT)`,
`(HTTHH)`,
`(HHTTH)`,
`(TTHHT)`,
`(TTHTH)`,
`(HHTHT)`,
`(HTHTT)`,
`(THTHH)`,
`(THTTH)`,
`(HTHHT)`,
`(HTHTT)`,
`(THTHH)`,
`(HHHTH)`,
`(TTTHT)`

Next, we list out all outcomes with at most 3 heads

, ,
`(HHHTT)`,
`(HHTTT)`,
`(HTTTT)`,
`(TTTTT)`,
`(TTTTH)`,
`(TTTHH)`,
`(TTHHH)`, ,
`(HTHTH)`,
`(THTHT)`, ,
`(TTHTT)`,
`(HTTHT)`,
`(THHTH)`,
`(THHHT)`,
`(HTTTH)`,
`(THHTT)`,
`(HTTHH)`,
`(HHTTH)`,
`(TTHHT)`,
`(TTHTH)`,
`(HHTHT)`,
`(HTHTT)`,
`(THTHH)`,
`(THTTH)`,
`(HTHHT)`,
`(HTHTT)`,
`(THTHH)`,
`(TTTHT)`

So, the number of set is:

`Sets = 27`