maths

(a)
No. of ways to get 5 cards randomly from 52 cards
= 52C5

To get one pair of "A" :
• Get 2 "A" out of the 4"A" (4C2).
• Then get 3 cards from the 48 cards that are NOT "A" (48C3).

The required probability
= 4C2 x 48C3 / 52C5
= (4! / 2!2!) x (48! / 3!45!) / (52! / 5!47!)
= 2162 / 54145


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(b)
To get two pairs :
• Choose 2 kinds from the 13 kinds (13C2).
• Get 2 cards from the 4 cards of each of the 2 kinds ((4C2)²)chosen above.
• Get 1 cards from the 44 cards of the other 11 kinds (44C1).

The required probability
= 13C2 x (4C2)² x 44C1/ 52C5
= (13! / 2!11!) x (4! / 2!2!)² x (44! / 1!43!) / (52! / 5!47!)
= 198 / 4165


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(c)
To get 3"A" and 1 pair :
• Get 3 "A" out of the 4 "A" (4C3).
• Choose 1 kind from the 12 kinds apart from "A" (12C1).
• Get 2 cards from the 4 cards of the kind chosen (4C2).

The required probability
= 4C3 x 12C1 x 4C2/ 52C5
= (4! / 3!1!) x (12! / 1!11!) x (4! / 2!2!) / (52! / 5!47!)
= 6 / 54145


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(d)
To get 4 of a same kind :
• To choose 1 kind from the 13 kinds (13C1).
• Get all 4 cards of the kind chosen (1).
• Get 1 card from the 48 cards of the rest 12 kinds (48C1).

The required probability
= 13C1 x 48C1 / 52C5
= (13! / 1!12!) x (48! / 1!47!) / (52! / 5!47!)
= 1 / 4165