A "hand" consists of a set of 5 cards selected from an ordinary deck of 52 cards:

(1) How many different hands contain exactly 4 diamonds? = 27885

(2) How many different hands contain exactly 4 cards of the same suit (any of the four suits)?

(3) How many different hands contain cards of exactly one suit?

Can someone help me figure out the last two questions?

so you want 4 out of the 13 diamonds ---> C(13,4)

then one more card from the 39 non-diamonds --> C(39,1)
so 715x39 = 27885

2. wouldn't the answer to 1. which was for diamonds be the same for the clubs, the same for the hearts, and the same for the spades ???
so 4x27885

3. I assume the suit would be specified.
If not, then for each specific suit is would be C(13,5) = 1287 , that is, all 5 cards come from the same 13 cards.

Thanks Reiny!!!

Certainly! I can help you with the last two questions.

(2) To determine the number of different hands that contain exactly 4 cards of the same suit, we can consider each suit separately. There are four suits in a standard deck of cards: hearts, diamonds, clubs, and spades.

For each suit, we can choose 4 cards from the 13 cards of that suit. This can be done in "13 choose 4" ways, which can be calculated using the formula:

C(n, r) = n! / (r!(n-r)!)

So, the number of different hands with exactly 4 cards of the same suit would be:

C(13, 4) x 4

Calculating this, we get:

C(13, 4) = 13! / (4!(13-4)!) = 715

Multiplying this by 4 (for each suit), we get:

715 x 4 = 2860

Therefore, there are 2860 different hands containing exactly 4 cards of the same suit.

(3) To find the number of different hands that contain cards of exactly one suit, we can again consider each suit separately.

For each suit, we need to choose all 5 cards from that suit. This can be done in "13 choose 5" ways, since there are 13 cards in each suit and we want to choose 5 of them.

Using the same formula as before, we can calculate:

C(13, 5) x 4

Calculating this, we get:

C(13, 5) = 13! / (5!(13-5)!) = 1287

Multiplying this by 4 (for each suit), we get:

1287 x 4 = 5148

Therefore, there are 5148 different hands containing cards of exactly one suit.

Sure! Let's go through each question step by step.

(2) How many different hands contain exactly 4 cards of the same suit (any of the four suits)?

To solve this question, we can break it down into smaller steps:

Step 1: Select a suit
There are 4 suits (hearts, diamonds, clubs, spades) to choose from. So, there are 4 ways to select a suit.

Step 2: Select 4 cards from the chosen suit
Within the selected suit, we need to choose 4 cards. Since each suit contains 13 cards in a standard deck, there are 13C4 ways to choose the 4 cards.

Step 3: Select 1 card from the remaining 3 suits
After selecting the 4 cards of the chosen suit, we need to select 1 additional card from the remaining 3 suits. Each of the remaining suits has 13 cards, so there are 13 ways to choose a card from each suit. Therefore, there are 13^3 ways to select 1 card from the remaining 3 suits.

Step 4: Calculate the total number of hands
To get the total number of hands, we multiply the results from each step together:
Total number of hands = (number of ways to select a suit) * (number of ways to choose 4 cards from the chosen suit) * (number of ways to choose 1 card from the remaining suits)

Total number of hands = 4 * 13C4 * 13^3 = 685,464

So, there are 685,464 different hands that contain exactly 4 cards of the same suit.

(3) How many different hands contain cards of exactly one suit?

To solve this question, we can use a similar approach as in question (2):

Step 1: Select a suit
There are 4 suits (hearts, diamonds, clubs, spades) to choose from. So, there are 4 ways to select a suit.

Step 2: Select 5 cards from the chosen suit
Since we want all cards in the hand to be of the same suit, we need to select all 5 cards from the chosen suit. Each suit contains 13 cards, so there are 13C5 ways to choose 5 cards from the chosen suit.

Step 3: Calculate the total number of hands
To get the total number of hands, we multiply the results from each step together:
Total number of hands = (number of ways to select a suit) * (number of ways to choose 5 cards from the chosen suit)

Total number of hands = 4 * 13C5 = 5148

So, there are 5148 different hands that contain cards of exactly one suit.

I hope this explanation helps! Let me know if you have any further questions.