In a certain card game, each player is initially dealt a hand consisting of 5 cards from a...
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In a certain card game, each player is initially dealt a hand consisting of 5 cards from a deck of 52 cards. The 52 cards deck consist of 4 suits (spades, hearts, clubs, and diamonds) and 13 ranks in each suit. How many ways you can deal a hand that contain at least one card from every suite?
Show answer & explanation
With 5 cards and 4 suits, one suit has 2 cards and the other three suits have 1 card each. Ways = C(4,1) * C(13,2) * 13^3 = 4 * 78 * 13^3 = 24 * 13^4 = (13^4) * 3 * (2^3).
Step-by-step Derivation:
Step 1: To have 5 cards across 4 suits with at least 1 per suit, the distribution must be 2, 1, 1, 1.
Step 2: Choose the suit with 2 cards: C(4,1) = 4.
Step 3: Choose 2 cards from that suit: C(13,2) = 78.
Step 4: Choose 1 card from each of the other 3 suits: 13^3.
Step 5: Total = 4 * 78 * 13^3 = 4 * (6 * 13) * 13^3 = 24 * 13^4 = 3 * 8 * 13^4 = (13^4) * 3 * (2^3).