Winner2013 wrote:Q. A pod of 6 dolphins always swims single file, with 3 females at the front and 3
males in the rear. In how many different arrangements can the dolphins swim?
source - MGMAT strategy guide
answer is - 36
I want to know where am i going wrong in my logic. Can someone help?
My logic :
6!
/ 3! * 3!
where am i going wrong?
thanks,
Your solution would be correct for the following problem:
How many ways can the letters AAABBB be arranged?
Number of ways to arrange 6 distinct elements = 6!.
But the arrangement here includes 3 identical A's and 3 identical B's.
To account for the identical elements, we must divide by the number of ways each set of identical elements can be arranged.
The reason:
When the identical elements swap places, the arrangement doesn't change.
Thus, the number of ways to arrange AAABBB = 6!/(3!3!) = 20.
We divide by 3! to account for the 3 identical A's and by another 3! to account for the 3 identical B's.
In the posted problem:
The 3 female dolphins must occupy the 3 front positions, while the 3 male dolphins must occupy the 3 back positions.
In the 3 front positions, the number of ways to arrange the 3 female dolphins = 3!.
In the 3 back positions, the number of ways to arrange the 3 male dolphins = 3!.
To combine these options, we multiply:
3! * 3! = 36.
Since the dolphins are not identical, no division is necessary.
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