# Math Help - Swimmers Medley (Combinations)

1. ## Swimmers Medley (Combinations)

Here is a question I have a problem with - I don't know how to find all ansers given restrictions.

Tayah, Innika, Jessica and Katherine have been selected for a swimming competition. Their coach sets the order they race in, following the conditions.

i] If Katherine starts, Jessica must go 3rd
ii] If Tayah or Katherine goes 2nd, Jessica must start.
iii] Innika runs 2nd if and only if Tayah runs third.
iv] If Katherine runs 3rd, then Tayah must run second.
v] Should Katherine run last, then Jessica must run third and Tayah starts.

List all possible orders which follow the conditions, giving explanations for the solutions.

Please explain how I should work towards the answer.

Thanks

2. Originally Posted by jt1894
Here is a question I have a problem with - I don't know how to find all ansers given restrictions.

Tayah, Innika, Jessica and Katherine have been selected for a swimming competition. Their coach sets the order they race in, following the conditions.

i] If Katherine starts, Jessica must go 3rd
ii] If Tayah or Katherine goes 2nd, Jessica must start.
iii] Innika runs 2nd if and only if Tayah runs third.
iv] If Katherine runs 3rd, then Tayah must run second.
v] Should Katherine run last, then Jessica must run third and Tayah starts.

List all possible orders which follow the conditions, giving explanations for the solutions.

Please explain how I should work towards the answer.

Thanks
Well the easiest way to go about it would be to list all possible orders without restrictions...
I J K T
I J T K
I T J K
I T K J
I K T J
I K J T

J K I T
J K T I
J T I K
J T K I
J I T K
J I K T

K I T J
K I J T
K J I T
K J T I
K T I J
K T J I

T I J K
T I K J
T K I J
T K J I
T J I K
T J K I

Then take, say the first condition and cross out all the ones that contradict it...
(I don't know how to do the 'spoiler' thing, so I'll just write really small so you can't see it without quoting - assuming you might rather work it out yourself..

I J K T
I J T K
I T J K
I T K J
I K T J
I K J T

J K I T
J K T I
J T I K
J T K I
J I T K
J I K T

K I J T
K T J I

T I J K
T I K J
T K I J
T K J I
T J I K
T J K I

Then the ones that contradict the second condition...

I J K T
I J T K

J K I T
J K T I
J T I K
J T K I
J I T K
J I K T

K I J T

T I J K
T I K J
T J I K
T J K I

Then the third...

I J K T
I J T K

J K I T
J T I K
J T K I
J I T K

T J I K
T J K I

Then the fourth

I J T K

J K I T
J T I K
J T K I
J I T K

T J I K

And finally..

J K I T
J T K I
J I T K

No guarantee that's correct, mind...

As for reasons...
They're the only ones that satisfy all conditions..?

Or you could just write out all the possibilities, and mark which number condition they contradict...

3. thanks a lot

yes you were right, i did want to work it out myself, and your 'whited out' answer was also very useful for me to not 'accidentally' cheat.

I will try work it out myself and compare with yours.

Thanks again