Final answer:
To calculate the number of possible 5-card hands meeting specific criteria, we use combinations. For each scenario, we select the required cards from the appropriate subsets and multiply the number of ways these selections can be made.
Step-by-step explanation:
To determine how many possible 5-card hands can be drawn from a standard 52-card deck that meet specific conditions, we must use combinations. A combination is a way of selecting items from a collection, such that the order of selection does not matter.
Three Spades and Two Non-Spades
We first choose three spades from the 13 available ones: this can be done in C(13,3) ways. Then, we choose two non-spades from the 39 non-spade cards: this can be done in C(39,2) ways. The total number of possible hands is the product of these two combinations, which is C(13,3) × C(39,2).
Four Face Cards and One Non-Face Card
Since there are 12 face cards (J, Q, K of each suit), we can choose four of them in C(12,4) ways. We then choose one non-face card from the remaining 40 cards (52 total cards minus 12 face cards) in C(40,1) ways. Multiply these combinations to find the total number of hands: C(12,4) × C(40,1).
Three Black Cards, One Heart, and One Diamond
We select three black cards from the 26 available (13 spades and 13 clubs) in C(26,3) ways. Then, we choose one heart from the 13 hearts in C(13,1) ways and one diamond from the 13 diamonds in the same way, C(13,1). The total for this scenario is C(26,3) × C(13,1) × C(13,1).