a) Find the number of ways to put ten pairs of socks into four drawers if each pair is a different color and both members of a pair do not have to go into the same drawer.

b) Change the socks to gloves. They are all the same color, but each pair contains a left and right glove.


closed as off-topic by JMoravitz, Trevor Gunn, Daniel W. Farlow, jvdhooft, Lazy Lee Jul 13 '17 at 5:31

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  • 2
    $\begingroup$ what has been tried ? $\endgroup$ – user451844 Jul 13 '17 at 1:45
  • $\begingroup$ For (a), pick how to distribute the red socks (they can either go in different drawers [which drawers?] or the same drawer [which drawer?]), and then given a distribution for the red socks, do the same for the blue socks, then the green socks, etc... apply multiplication principle. For (b), first choose how to distribute the leftgloves, then the right gloves. Apply stars-and-bars and then multiplication principle. $\endgroup$ – JMoravitz Jul 13 '17 at 2:06


There are 10 pairs of socks, independently assigned drawers.

Each pair of sock can be assigned drawers in how many distinguishable ways? (Hint: triangular numbers)

There are $(\text{that many})^{10}$ ways to make that many choices for 10 pairs of socks.


There are now 4 drawers to assign independently to 10 indistinct objects of one category and then 10 indistinct objects of a second category. (Hint: stars-and-bars)


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