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

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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) About Me  (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) About Me  (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?