SOLUTION: a standard deck of cards has 52 cards composed of 13 hearts, 13 diamonds, 13 spades, and 13 clubs. in how many ways can hands composed of 5 cards be drawn if they must contain exac

Question 1192276: a standard deck of cards has 52 cards composed of 13 hearts, 13 diamonds, 13 spades, and 13 clubs. in how many ways can hands composed of 5 cards be drawn if they must contain exactly three diamonds?
Found 2 solutions by math_tutor2020, Alan3354:
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

n = 13 diamonds
r = 3 diamonds to select

Order doesn't matter with card hands.
Use the nCr combination formula.
n C r = (n!)/(r!(n-r)!)
13 C 3 = (13!)/(3!*(13-3)!)
13 C 3 = (13!)/(3!*10!)
13 C 3 = (13*12*11*10!)/(3!*10!)
13 C 3 = (13*12*11)/(3!)
13 C 3 = (13*12*11)/(3*2*1)
13 C 3 = (1716)/(6)
13 C 3 = 286
There are 286 ways to pick the three diamond cards in any order.

The remaining n = 52-13 = 39 non-diamond cards are then picked from to select r = 2 more cards.
Plug n = 39 and r = 2 into the nCr formula
n C r = (n!)/(r!(n-r)!)
39 C 2 = (39!)/(2!*(39-2)!)
39 C 2 = (39!)/(2!*37!)
39 C 2 = (39*38*37!)/(2!*37!)
39 C 2 = (39*38)/(2!)
39 C 2 = (39*38)/(2*1)
39 C 2 = (1482)/(2)
39 C 2 = 741
There are 741 ways to pick the other two non-diamond cards.

We have 286*741 = 211,926 ways to form a five card hand of exactly three diamonds.

Answer by Alan3354(69443) (Show Source): You can put this solution on YOUR website!
How many people on Earth don't know the composition of a standard deck of cards?
RELATED QUESTIONS
Consider a standard deck of 52 cards having 13 clubs, 13 diamonds, 13 hearts, and 13... (answered by galactus)
How many ways can you deal a hand of 13 playing cards that contain 4 spades, 7 diamonds,... (answered by greenestamps)
A single card is drawn from a standard 52-deck of cards with four suits: hearts, clubs,... (answered by Edwin McCravy)
A standard deck of cards consists of four suits (clubs, diamonds, hearts, and spades),... (answered by math_tutor2020,greenestamps)
A standard deck of cards consists of four suits (clubs, diamonds, hearts, and spades),... (answered by ikleyn,Edwin McCravy)
A standard deck of cards has four suits (hearts, diamonds, spades, and clubs) each with... (answered by Boreal)
A standard deck of cards has four suits: hearts, diamonds, clubs, and spades. Each suit (answered by stanbon)
A standard deck of 52 playing cards will have cards in fours suits (clubs, diamonds,... (answered by checkley79)
There are 52 cards in the standard deck of cards. 26 of them are red: 13 Hearts �, 13... (answered by ewatrrr)