You pick cards one at a time without replacement from an ordinary deck of 52 playing cards. what is the minimum number of cards you must pick in order to guarantee that you geta)two pair (for example, two kings or two 5s)b)three of a kind (for example, three 7s)

Part A)

There are 52 cards total. There are four suits with each suit having the same value labels (eg: each suit has a '4' card). If you select 13 cards, then there is a chance you got a pair in there somewhere. However, there's also a chance that NONE of the cards are repeats. So it's possible, though very unlikely, to pick out a full suit with no repeats.

If that unlikely scenario happens then you just need to pick one more card to guarantee you have one pair. That 14th card will be a repeat of some other card just of a different suit (eg: 2 of hearts and 2 of spades)

Final 14 cards



Part B)

Using the same logic of part A, we can pick out 26 cards and have each card be repeated (from 2 all the way up to ace). This is very unlikely not to have a three of a kind somewhere in there. If there are no three of a kinds, then simply pick another card to guarantee that one set of 3 cards are the same. This is effectively using the pigeon hole principle. 

27 cards

Hello! i hope this
since the cosine is the ratio of the adjacent side to the hypotenuse, you can see that cos 60° = ½. answer. according to the property of cofunctions (topic 3), sin 30° is equal to cos 60°. sin 30° = ½.