S. How many hands of five playing cards (from a standard deck) satisfy the follo

Question

S. How many hands of five playing cards (from a standard deck) satisfy the following conditions? (Make sure to show your work. Each factor or term in your calculation should be explained.) a. The h and has up to two aces and no other cards of matching rank. (For examp four other cards all of different ranks from one another. Suits do not matter.) le, (AH, 2C, 3C, 4D, 7Hj would be an allowable hand since it has one ace and b. The hand has exactly two kings or exactly two queens. (This is an "inclusive or. There are no restriction on the other cards.)

Explanation / Answer

Hi,
A. Lets split this into 2 parts
0. 0 Aces and different ranks
5C1*48C1*44C1*40C1*36C1*32C1 (choosing among the five cards * choosing any card except aces and then subsequent card will have 4 less choices since we cant choose the same rank)
1. Only one Ace and different ranks
5C1 * 4C1 * 48C1*44C1*40C1*36C1  (choosing 1 among the five cards * choosing 1 Ace among 4 and then subsequent card will have 4 less choices since we cant choose the same rank)
2. 2 Aces and dfferent ranks
  5C2 * 4C2 * 48C1*44C1*40C1(choosing 2 among the five cards * choosing 2 Ace among 4 and then subsequent card will have 4 less choices since we cant choose the same rank)
Total woud be sum of 0+1+2
Thumbs up if this was helpful, otherwise let me know in comments

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