Question 1192276
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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 = <font color=red>211,926</font> ways to form a five card hand of exactly three diamonds.
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