# How Many Ways to Get Three of a Kind in a Four-Card Hand?

• blue_soda025
In summary, there are 2496 possible combinations to obtain three of a kind in a game of cards using only four cards from a standard deck of 52 cards.
blue_soda025
Suppose you play a game of cards in which only four cards are dealt from a standard deck of 52 cards. How many ways are there to obtain three of a kind? (3 cards of the same rank and 1 card of a different rank, for example 3 tens and 1 queen.)

Could someone help me with how to do this problem? I tried doing 4C3 x 13 x 52C1, which was obviously wrong. :/ Any help would be appreciated.

You can choose first card in 52 different ways. You can choose the next two (same of a kind as the first one) in 3 * 2 ways. And the last card, in 49 different ways.

blue_soda025 said:
Suppose you play a game of cards in which only four cards are dealt from a standard deck of 52 cards. How many ways are there to obtain three of a kind? (3 cards of the same rank and 1 card of a different rank, for example 3 tens and 1 queen.)

Could someone help me with how to do this problem? I tried doing 4C3 x 13 x 52C1, which was obviously wrong. :/ Any help would be appreciated.
The standard 52 cards contain 13 different ranks of 4 cards each.
For any given rank, there are C(4,3) combinations of 3 cards chosen from the rank's 4 cards. Since there are 13 different ranks, a total of {13*C(4,3)} possible combinations of {3 cards from the same rank} exist. Finally, there remain {(52 - 4) = 48} cards in the 12 other (different) ranks from which to choose the final card. Hence:
{Total Combinations of "3-of-a-Kind" from std 52 Cards} = {13*C(4,3)}*(48) = (2496)

~~

## What is combinatorics?

Combinatorics is a branch of mathematics that focuses on counting and arranging objects, typically without replacement and in a specific order. It involves the study of patterns and structures in discrete objects, such as graphs, permutations, and combinations.

## What types of problems can be solved using combinatorics?

Combinatorics can be applied to a wide range of problems, including those related to probability, coding theory, graph theory, and optimization. It can also be used to solve problems in various fields such as computer science, physics, and biology.

## What are some common techniques used in combinatorics?

Some common techniques used in combinatorics include the fundamental principle of counting, combinations, permutations, and generating functions. Other techniques include the pigeonhole principle, inclusion-exclusion principle, and recurrence relations.

## What is the difference between combinations and permutations?

Combinations and permutations are both ways of selecting objects from a set, but they differ in terms of order. In combinations, the order of selection does not matter, whereas in permutations, the order does matter. For example, selecting three out of five objects without replacement can be done in 10 different ways using combinations, but in 60 different ways using permutations.

## How is combinatorics used in real-life situations?

Combinatorics has many real-life applications, such as calculating the number of possible outcomes in a game of chance, optimizing travel routes, and designing efficient coding schemes. It is also used in fields like genetics, where it can help determine the number of possible genetic combinations in offspring.

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