Answer:
10.55% probability
Step-by-step explanation:
A probability is the number of desired outcomes divided by the number of total outcomes.
The order in which the CDs are chosen is not important. So we use the combinations formula to solve this question.
1 Bach CD, from a set of 4.
1 Beethoven CD, from a set of 6.
1 Brahms CD, from a set of 3.
1 Handel CD, from a set of 2.
So, D=144
4 CDs from a set of 4+6+3+2 = 15.
So, T= 1365
p= D/T= 144/1365 = 0.1055
10.55% probability that she will choose one by each composer
First, calculate for the volume of the cube before each edges are cut.
V = e³
where e is the length of each sides. Substituting the known value,
V = (4/5 cm)³ = 0.512 cm³
Then, calculate for the volume of each of the small cubes cut out from the corners.
V = (1/5 cm)³ = 0.008 cm³
Since there are 8 of these small cube, we multiply the volume by 8.
8V = 8(0.008 cm³) = 0.064 cm³
Then, subtracting the volumes will give us an answer of <em>0.448 cm³</em>
<span>Assuming that "pair up students" means "divide up all 20 of the students into groups of two," and we regard two pairings as the same if and only if, in each pairing, each student has the same buddy, then I believe that your answer of 20! / [(2!)^10 * (10!)] is correct. (And I also believe that this is the best interpretation of the problem as you've stated it.)
There are at least two ways to see this (possibly more).
One way is to note that, first, we have to select 2 students for the first pair; that's C(20, 2) (where by C(20, 2) I mean "20 choose 2"; that is, 20! / (18! * 2!). )
Then, for each way of selecting 2 students for the first pair, I have to select 2 of the remaining 18 students for the second pair, so I multiply by C(18, 2).
Continuing in this manner, I get C(20, 2) * C(18, 2) * ... * C(2, 2).
But it doesn't matter in this situation the order in which I pick the pairs of students. Since there are 10! different orders in which I could pick the individual pairs, then I want to divide the above by 10!, giving me the answer
[C(20, 2) * C(18, 2) * ... * C(2, 2)] / 10!.
This is the same as your answer, because C(n , 2) = n(n - 1) / 2, so we can simplify the above as
[(20 * 19) / 2 * (18 * 17) / 2 * ... * (2 * 1) / 2] / 10!
= 20! / [2^10 * 10!]
= 20! / [(2!)^10 * (10!)].
Another way is to reason as follows:
1. First, arrange the 20 students in a line; there are 20! ways to do this
2. We can get a pairing by pairing the 1st and 2nd students in line together, the 3rd and 4th students together, etc.
3. But if I switch the order of the 1st and 2nd student, then this doesn't give a different pairing. I don't want to count both orderings separately, so I divide by 2!
4. The same argument from step 3 holds for the 3rd and 4th student, the 5th and 6th student, etc., so I need to divide by 2! nine more times
5. Finally, the particular order in which I selected the ten pairings are unimportant--for example, the following orderings don't produce different pairings:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20
3, 4, 1, 2, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20
So I need to further divide by the number of ways I can arrange the ten pairs, which is 10!.
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