sweetkendra said:
There are 7 seals and 8 dolphins entered in an aquatic obedience show. How many ways could you select a final group to judge if it must consist of 3 seals and 3 dolphins.
You have six slots to fill. Order doesn't matter but, for convenience, let's say that the first three slots will be filled with seals and the last three with dolphins. So you have six slots to fill:
. . . . .___ * ___ * ___ * ___ * ___ * ___
You have seven choices for the first slot:
. . . . ._
7_ * ___ * ___ * ___ * ___ * ___
Once you have picked one seal, you have six choices remaining for the second slot, and then five choices for the third slot:
. . . . ._
7_ * _
6_ * _
5_ * ___ * ___ * ___
By similar reasoning, you have eight choices for the fourth slot, seven for the fifth, and six for the sixth:
. . . . ._
7_ * _
6_ * _
5_ * _
8_ * _
7_ * _
6_
Multiplying, you get 70560 ways of choosing the animals in the order SSSDDD (where "S" stands for a seal and "D" stands for a dolphin). But this ordering is not necessary so, to get rid of the permutation-type selection I've done above, we need to divide by 3! 3! (one "three factorial" for each of the two permuted sets, to "divide out" the implied ordering, which we don't need). This will give you an answer which matches what you
should have gotten with the combination-formula below.
sweetkendra said:
nPr= n!/(n-r)!
nCr= n!/(n-r)!r!
7C3 * 8C3= 7!/(7-3)!3! * 8!/(8-3)!3! = 7!/4!3! * 8!/5!3! = 7*6*5/4*3*2 * 8*7*6/5*4*3 = 210/24 * 336/60 = 49
First, let me say "Thank you!" for defining your terms and showing your work! You make a tutor's work so much easier!
Looking at your last line above, I think perhaps some factors were dropped...?
. . . . .\(\displaystyle \left(\frac{7!}{3!\,4!}\right)\,\left( \frac{8!}{5!\,3!}\right)\, =\, \left(\frac{7\times 6\times 5}{3\times 2\times 1}\right)\,\left( \frac{8\times 7\times 6}{3\times 2\times 1}\right)\)
. . . . . . . . . . . . . . . . . .. . .\(\displaystyle =\, \left( \frac{7\times 5}{1} \right)\, \left( \frac{8\times 7}{1}\right)\)
...which does not equal 49. :shock:
Note that, with the arithmetic corrected, your method and the slot-filling method displayed above both give the same answer.
Eliz.
