Poker Five cards are chosen from an ordinary deck to form a...

Poker Five cards are chosen from an ordinary deck to form a hand in poker. In how many ways is it possible to get the following results?
a. 4 queens b. No face card
c. Exactly 2 face cards d. At least 2 face cards
e. 1 heart, 2 diamonds, and 2 clubs

Key Concepts

Combination Formula

This concept involves calculating the number of ways to choose k objects from a set of n distinct objects when the order of selection does not matter. It is often represented by the binomial coefficient, n choose k, and it is a fundamental tool in solving problems related to counting subsets, such as card hands in poker.

Counting with Restrictions

This concept deals with solving counting problems where certain conditions or constraints are imposed. For example, selecting items that must belong to specific categories or selecting an exact number of particular types requires adjusting the basic combination formulas to account for these restrictions.

Multiplication Principle

This principle states that if one event can occur in m ways and a second, independent event can occur in n ways, then the two events together can occur in m times n ways. It is used to combine separate independent choices or stages in a counting problem, such as selecting cards from different suits.

Complement Principle

This concept is used to simplify counting problems, particularly those involving 'at least' or 'at most' restrictions. Instead of directly counting the desired outcomes, it can be easier to count the total number of outcomes and subtract the number of unwanted outcomes, thereby arriving at the count for the desired event.

Transcript

00:01 Okay, this question is going to ask us about the number of different ways we could get certain poker hands.

00:08 And really, we just need to know that a poker hand is formed from any five cards in a normal deck.

00:15 So the first one asks, how many ways could we get a poker hand with four queens in it? so, this is equal to the number of ways to get the queens times the number of ways to get everything else.

00:29 So with the four queens, there are four queens in the deck, one pursuit.

00:41 So four queens, and we're picking four of them, times the choices for the last card.

00:49 So there's 52 cards in a deck, which means that for the non -que queens, we have 48 to pick from, and we're choosing one.

00:57 So this is just 48.

00:59 Then we want no face card.

01:09 So we need to figure out the number of face cards so we know how many we can avoid.

01:14 So for the face cards, we have a jack, a queen, and a king, and each of these there are four, one pursuit, because we have four jacks, four queens, and four kings.

01:35 So this means that we have 40 cards to pick from, and we want all five to come from there.

01:47 So our answer is just 40 choose five, because we don't want any of the 12 face cards, and plugging this in, 40 choose five is 658 ,000, and 8.

02:08 Then it wants exactly two face cards.

02:13 So we have the number of ways to pick the face card, times the number of ways to pick the face card, pick the non -face cards.

02:21 So we said over here that there are 12 total face cards, and we want two of them, times the remaining 40 cards where we're going to pick three of them...

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