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Number of ways of selecting 5 cards from a deck of 52 cards such that one card of each suit is?
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Number of ways of selecting 5 cards from a deck of 52 cards such that ...
For a five-card hand to have one card in each suit, it must be the case that the hand contains two cards from one of the suits and one card from each of the others. For the suit with two cards, there are (132) ways of choosing those two cards. For each of the other 3 suits, there are 13 ways of choosing a card from that suit. Then, because there are 4 ways of choosing which suit contributes 2 cards to the hand, we obtain the following result:
(132)x13x13x13x4
This simplifies to 685464.
The issue with the 1370928 answer is that it double counts each hand. Consider a hand with both the 2 of hearts and the 3 of hearts. It could be the case that we select the 2 of hearts as the "representative" (one of 13) for its suit, and the 3 of hearts as the final card from the 48 remaining at the end. Alternatively, though, we could select the 3 of hearts as the "representative" for its suit, and the 2 of hearts as the final card. Both of these cases yield the same hand, though. In this manner, then, every hand is double counted. So, the 13 x 13 x 13 x 13 x 48 approach is close, but you'd need to divide by 2 to account for this double-counting and arrive at the right answer.
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Number of ways of selecting 5 cards from a deck of 52 cards such that ...
The Problem:
We need to determine the number of ways to select 5 cards from a standard deck of 52 cards such that one card of each suit is included in the selection.
Solution:
To solve this problem, we can break it down into four cases, one for each suit (hearts, diamonds, clubs, and spades). We will calculate the number of ways to select a card from each suit and then multiply these counts together to get the total number of ways.
Case 1: Selecting a card from the hearts suit:
There are 13 cards in the hearts suit, and we need to select 1 card from this suit. Therefore, there are C(13,1) = 13 ways to choose a card from the hearts suit.
Case 2: Selecting a card from the diamonds suit:
Similar to the previous case, there are 13 cards in the diamonds suit, and we need to select 1 card from this suit. Therefore, there are C(13,1) = 13 ways to choose a card from the diamonds suit.
Case 3: Selecting a card from the clubs suit:
Again, there are 13 cards in the clubs suit, and we need to select 1 card from this suit. Therefore, there are C(13,1) = 13 ways to choose a card from the clubs suit.
Case 4: Selecting a card from the spades suit:
Similarly, there are 13 cards in the spades suit, and we need to select 1 card from this suit. Therefore, there are C(13,1) = 13 ways to choose a card from the spades suit.
Total number of ways:
Since we need to select one card from each of the four suits, we can multiply the counts from each case to get the total number of ways. Therefore, the total number of ways to select 5 cards such that one card of each suit is included is:
13 * 13 * 13 * 13 = 28,561
Conclusion:
There are 28,561 ways to select 5 cards from a deck of 52 cards such that one card of each suit is included in the selection.
We need to determine the number of ways to select 5 cards from a standard deck of 52 cards such that one card of each suit is included in the selection.
Solution:
To solve this problem, we can break it down into four cases, one for each suit (hearts, diamonds, clubs, and spades). We will calculate the number of ways to select a card from each suit and then multiply these counts together to get the total number of ways.
Case 1: Selecting a card from the hearts suit:
There are 13 cards in the hearts suit, and we need to select 1 card from this suit. Therefore, there are C(13,1) = 13 ways to choose a card from the hearts suit.
Case 2: Selecting a card from the diamonds suit:
Similar to the previous case, there are 13 cards in the diamonds suit, and we need to select 1 card from this suit. Therefore, there are C(13,1) = 13 ways to choose a card from the diamonds suit.
Case 3: Selecting a card from the clubs suit:
Again, there are 13 cards in the clubs suit, and we need to select 1 card from this suit. Therefore, there are C(13,1) = 13 ways to choose a card from the clubs suit.
Case 4: Selecting a card from the spades suit:
Similarly, there are 13 cards in the spades suit, and we need to select 1 card from this suit. Therefore, there are C(13,1) = 13 ways to choose a card from the spades suit.
Total number of ways:
Since we need to select one card from each of the four suits, we can multiply the counts from each case to get the total number of ways. Therefore, the total number of ways to select 5 cards such that one card of each suit is included is:
13 * 13 * 13 * 13 = 28,561
Conclusion:
There are 28,561 ways to select 5 cards from a deck of 52 cards such that one card of each suit is included in the selection.
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Number of ways of selecting 5 cards from a deck of 52 cards such that one card of each suit is?
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Number of ways of selecting 5 cards from a deck of 52 cards such that one card of each suit is? for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about Number of ways of selecting 5 cards from a deck of 52 cards such that one card of each suit is? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Number of ways of selecting 5 cards from a deck of 52 cards such that one card of each suit is?.
Number of ways of selecting 5 cards from a deck of 52 cards such that one card of each suit is? for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about Number of ways of selecting 5 cards from a deck of 52 cards such that one card of each suit is? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Number of ways of selecting 5 cards from a deck of 52 cards such that one card of each suit is?.
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