Question - A "standard" deck of playing cards consists of 52 cards in each of the 4 suits of Spades, Hearts, Diamonds, and Clubs. Each suit contains 13 cards: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King. In how many ways 5 cards can be selected from a "standard" deck of playing cards, without a replacement, so that all 4 suits appear?
- 1,287
- 4,056
- 52,728
- 405,646
- 685,464
“A "standard" deck of playing cards consists of 52 cards in each of the 4 suits of Spades, Hearts, Diamonds, and Clubs” – is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT by Mathivanan Palraj". GMAT Quant section consists of a total of 31 questions. To solve GMAT Problem Solving questions a student must have knowledge about a good number of qualitative skills. The GMAT quant topics in the problem-solving part require calculative mathematical problems that should be solved with proper mathematical knowledge.
Solution and Explanation:
Approach 1:
Given:
- A "standard" deck of playing cards consists of 52
- Each of the 4 suits of Spades, Hearts, Diamonds, and Clubs
- Each suit contains 13 cards: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King.
Conditions:
- All 4 suits should appear
Find Out:
- In how many ways 5 cards can be selected from a "standard" deck of playing cards, without a replacement,
We know that the formula for selecting 1 item from n items = nC1
Step 1: There will be 1 suit out of 4 in which 2 cards will be chosen
Step 2: The number of ways to select that suit = 4C1 = 4
Step 3: The number of ways 5 cards can be selected from the deck of playing cards, without a replacement, so that all 4 suits appear:
=> 4* 13C1*13C1*13C1*13C2
=> 685,464
Hence, E is the correct answer.
Approach 2:
Given:
- A "standard" deck of playing cards consists of 52
- Each of the 4 suits of Spades, Hearts, Diamonds, and Clubs
- Each suit contains 13 cards: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King.
Conditions:
- All 4 suits should appear
Find Out:
- In how many ways 5 cards can be selected from a "standard" deck of playing cards, without a replacement
Conditions that we must keep in mind:
- Each suit is represented in the 5 cards
- Cards are to be replaced
Since the target is to make sure that all the suits appear, we have to consider each of them.
Ways in which 5 cards can appear are :
2S H D C
S 2H D C
S H 2D C
S H D 2C
So there are 4 ways in total.
As cards had to be replaced at any moment there are 52 cards in the deck i.e. 13 cards in each suit.
So, required answer is
=> 4* 13C2*13C1*13C1*13C1
=> 685,464
Hence, E is the correct answer.
Approach 3:
There gonna be 1 suit of which there will be 2 cards appear.
Number of ways to select that suit = 4C1 = 4
Number of ways 5 cards can be selected from a "standard" deck of playing cards, without a replacement, so that all 4 suits appear
= 4* 13C1*13C1*13C1*13C2
= 685464
Hence, E is the correct answer.
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