Question 1197057: Hello! Please help me with this question:
There are 10 students with different designs of gloves. A professor asked them to put all their gloves in a box and he randomly selected 6 pieces of gloves. What is the probability that there is exactly one matching pair of gloves?
My initial solution to this problem is 6/60 = 1/10. But I'm not entirely sure about this. Can anyone help me? Thank you!
Found 3 solutions by greenestamps, ikleyn, ewatrrr:
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
It is hard for anybody to help you when the problem is not described clearly....
"... 10 students with different designs of gloves."
That probably mean 10 different pairs of gloves, making 20 gloves all together. But it might mean "gloves" for taking hot dishes out of the oven, which would make only 10 gloves.
"... 6 pieces of gloves."
That is very strange English. "Pieces of gloves" makes it seem that not all the gloves are intact.
And your initial solution "6/60" makes it appear as if there are 60 gloves all together, which doesn't make sense any way I can interpret the problem.
If the problem is 10 different pairs of gloves with the professor randomly selecting 6 of the 20 total, then write the problem that way.
But as it is, we have no idea what the problem really is.
Re-post clearly....
Answer by ikleyn(52781)  (Show Source):
You can put this solution on YOUR website! .
There are 10 students with different designs of gloves.
A professor asked them to put all their gloves in a box and he randomly selected
6 pieces of gloves. What is the probability that there is exactly one matching pair of gloves?
~~~~~~~~~~~~~~~~~
There are 20 separate pieces in the box (of 10 pairs of gloves).
The probability the problem asking for, is the ratio, whose denominator is the number of all
sextuples that can be selected from the set of 20 items,  = 38760.
The numerator is the number of all sextuples that have one matching pair and 4 non-matching items.
Our task is to calculate the number of such sextuples.
In turn, this number is the product of 10 (the number of gloves) multiplied by the number
of all possible quadruples that consist of non-matching pairs from remaining 2*(10-1) = 18 items.
Thus our task is to calculate the number of all non-matching quadruples of 18 separate pieces.
We can select 1st item as any of 18 pieces, which gives us 18 options.
After that, we can choose next item from remaining 18-2 = 16 items (excluding item #1 and its twin).
It gives us 16 possible options.
After that, we can choose 3rd item from remaining 16-2 = 14 items (excluding item #1 and its twin,
as well as item #2 and its twin). It gives us 14 possible options.
After that, we can choose 4th item from remaining 14-2 = 12 items (excluding item #1 and its twin,
item #2 and its twin, item #3 and its twin). It gives us 12 possible options.
So, the number of all different quadruples of non-matching items from the total of 18 items (9 gloves)
is 18*16*14*12 = 48384.
Notice that 48384 is the number of all such different  quadruples,
so we need divide this number by 4! = 24, to count  quadruples.
Thus, the number of unordered quadruples is 48384/24 = 2016,
and the number of all sextuples we are interesting to count is 10*2016 = 20160.
Now the probability we are looking for is P =  =  = 0.5201 (rounded). ANSWER
Solved.
Answer by ewatrrr(24785)  (Show Source):
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