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Although rare, tied swimming races occur with some frequency. In fact, at the 2016 Olympics there was a three-way tie for second position in the
men's 100 m butterfly event.
Find the number of different ways four swimmers can finish a race if
(a) there are no ties
(b) we include the possibility of ties.

Answer :

subtomex0

[tex]a) \: 4 \times 3 \times 2 \times 1 = 24 \: ways[/tex]

[tex]b) \: this \: one \: is \: a \: little \: more \: \\ complex \: so \: bear \: with \: me \\ poss \: 1 = all \: \: tie \: for \: first \\ poss \: 2 = 3 \: tie \: for \: second \\ poss \: 3 = 2 \: tie \: for \: third \\ poss \: 4 = no \: one \: ties[/tex]

They can all tie for first place in one way,

which is, that they all arrive first with no order.

Possibility 1 = 1 way

If one person arrives in first place, this leaves 3 to tie for second, All 4 players can arrive first and the other 3 would have to tie in second so,

Possibility 2 = 4 ways

If two people take the first two spots, this leaves two people to tie for third place, All four players can occupy the first position and the other 3 can switch in second leaving two to tie at last. and this can happen in 4 ways.

Possibility 3 = 4(3) = 12 ways

Possibility 4 = 24 ways (solved)

Total ways = 1 + 4 + 12 + 24 = 41 ways

Answer = 41 ways

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