SOLUTION: How many ways are there to choose two twos from a standard 52-card deck? How many ways are there to choose three cards from a standard 52-card deck without choosing any twos? H

Question 1192165: How many ways are there to choose two twos from a standard 52-card deck?
How many ways are there to choose three cards from a standard 52-card deck without choosing any twos?
How many five-card hands (drawn from a standard 52-card deck) contain exactly two twos?
How many five-card hands (drawn from a standard 52-card deck) contain a two-of-a-kind?

Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
Let's break down each part of the problem:
**1. Choosing Two Twos:**
* There are four twos in a standard 52-card deck.
* We want to choose two of them.
* The number of ways to do this is given by the combination formula "4 choose 2":
⁴C₂ = 4! / (2! * 2!) = (4 * 3) / (2 * 1) = 6
**2. Choosing Three Cards Without Any Twos:**
* There are 48 cards in the deck that are *not* twos (52 total cards - 4 twos).
* We want to choose three of these 48 cards.
* The number of ways to do this is:
⁴⁸C₃ = 48! / (3! * 45!) = (48 * 47 * 46) / (3 * 2 * 1) = 17296
**3. Five-Card Hands with Exactly Two Twos:**
* First, we choose two twos (as calculated in part 1): ⁶C₂ = 6 ways.
* Then, we need to choose the remaining three cards from the 48 non-two cards (as calculated in part 2): ⁴⁸C₃ = 17296 ways.
* To get the total number of five-card hands with exactly two twos, we multiply these two results together:
6 * 17296 = 103776
**4. Five-Card Hands with a Two-of-a-Kind:**
This is trickier because it includes more than just two twos. It means *any* pair (two of the same rank) but not three or four of a kind.
* **Choose the rank for the pair:** There are 13 ranks in a deck (Ace through King). We choose one of these ranks: ¹³C₁ = 13 ways.
* **Choose the two cards for the pair:** There are four cards of the chosen rank. We choose two of them: ⁴C₂ = 6 ways.
* **Choose the ranks for the other three cards:** We need three more cards, each of a different rank, and these ranks must be different from the rank we chose for the pair. There are 12 remaining ranks, and we choose three of them: ¹²C₃ = 220 ways.
* **Choose the suit for each of the three remaining cards:** For each of the three cards, we have 4 suit options. This gives us 4 * 4 * 4 = 4³ = 64 possibilities.
* **Multiply everything together:** 13 * 6 * 220 * 64 = 1098240
Therefore, there are 1,098,240 five-card hands with exactly one pair (two of a kind).

RELATED QUESTIONS
How many ways are there to choose one card from a standard 52-card deck without choosing... (answered by jim_thompson5910)

a. How many ways are there to choose four fives from a standard 52-card deck? b. How... (answered by KMST)

In a 52 card deck, how many ways are there to get 21 in two... (answered by Fombitz)

How do you properly use the combination formula,I'm confused on how to use the... (answered by ikleyn)

How many ways are there to choose five cards from 52 so that you have two cards with the (answered by Edwin McCravy)

In how many ways can three cards be selected from a 52-card deck if the selected cards... (answered by Alan3354)

How many ways are there to choose five cards from 52 so that you have four cards with... (answered by Edwin McCravy)

1.) How many different ways can 3 cards be drawn from a deck of 52 cards without... (answered by Boreal,ikleyn)

Two cards are drawn in succession, without replacement, from a standard deck of 52 cards. (answered by scott8148)