5 cards are drawn from a normal deck of cards.
a) What are the odds that drawn was 1 five card and 2 three cards?
b) What are the odds that 3 of the cards were of the same suit?
A standard deck of cards has 52 cards, 4 of each denomination. The probability that the first card drawn is a 5 is 4/52= 1/13. There are then 51 cards left, 4 of them 3s. The probability the second card drawn is a 3 is 4/51. There are then 50 cards left, three of them 3s. The probability the third card drawn is a 3 is 3/50. The last two cards can be any except a 5 or a 3. Since there are 3 5s and 2 3s left, there are 49- 5= 44 "non 3 or 5 cards" left. The probability the last two cards drawn are not a 3 or a 5 is (44/49)(43/48). The probability or one 5, two threes, and two other cards, in that order is (1/13)(4/51)(3/50)(44/49)(43/48).
But there are 5!/(1!2!2!) = 30 different orders of "533AA" (A is "any other" card). It is fairly easy to see that the probability for any specific order is the same.
The other problem is done the same way but is slightly easier.