SOLUTION: This exercise refers to a standard deck of playing cards. Assume that 5 cards are randomly chosen from the deck. How many hands contain exactly 3 jacks?

Question 1132949: This exercise refers to a standard deck of playing cards. Assume that 5 cards are randomly chosen from the deck.
How many hands contain exactly 3 jacks?
Found 2 solutions by math_helper, rothauserc:
Answer by math_helper(2461) (Show Source): You can put this solution on YOUR website!
Using notation nCr = n!/((n-r)!r!)

The number of three of kind hands (where three cards match and the other two do not match) is:

The factors are:
(selection of rank)*(#arrangments of three matching cards)*(selection of nonmatching two cards)*(number of arrangements of those two)

The number of full houses (three matching and other two cards match):

Thus the total number of hands where three cards match, regardless of the remaining two cards is 54912+3744 = 58656

But, the problem only asks about jacks, which means the above number must be divided by (13C1) since the rank is chosen for you:

58656/13 = 4512

---
The number of hands with exactly three jacks is
(This also answers the question about how many hands have exactly three queens, as the rank does not matter).
----
Thanks tutor rothauserc! Yes, that approach is much more direct! I had poker hands in mind and went the round-about way. For the student, the first factor in tutor rothauserc's solution is the number arrangments of 3 jacks out of 4, and the 2nd factor is the number of arrangements of the remaining 2 cards out of 48. What I like about this website is I get to learn too :-)

Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
A standard deck of playing cards consists of 52 cards
:
nCr = n! /(r! * (n-r)!)
:
4C3 * 48C2 = 4512 hands contain exactly 3 jacks
:

RELATED QUESTIONS
This exercise refers to a standard deck of playing cards. Assume that 5 cards are... (answered by Theo)
This exercise refers to a standard deck of playing cards. Assume that 5 cards are... (answered by Fombitz)
This exercise refers to a standard deck of playing cards. Assume that 5 cards are... (answered by Alan3354)
This exercise refers to a standard deck of playing cards. Assume that 5 cards are... (answered by Edwin McCravy)
This exercise refers to a standard deck of playing cards. Assume that 5 cards are... (answered by ikleyn)
This exercise refers to a standard deck of playing cards. Assume that 5 cards are... (answered by math_tutor2020)
This exercise refers to a standard deck of playing cards. Assume that 5 cards are... (answered by math_helper,math_tutor2020)
This exercise refers to a standard deck of playing cards. Assume that 5 cards are... (answered by Boreal)
This exercise refers to a standard deck of playing cards. Assume that 8 cards are... (answered by Edwin McCravy)