Question

How many combinations of 4 spades can be pulled from the 13 spades in a standard deck of cards?

Answer 1

Hint: The first spade can be chosen from the deck in 13 ways. The second spade can be chosen from the deck in 12 ways. The third spade can be chosen from the deck in 11 ways and the fourth spade can be chosen from the deck in 10 ways. The product of the ways in which 4 spades are chosen from the deck gives us the required combination.

Complete step by step solution:
The given question is solved using combinations. Combinations usually are the number of ways the objects can be selected without replacement.
Order is not usually a constraint in combinations, unlike permutations.
In the given question,
We are supposed to find the combinations of the event.
Event: 4 spades can be pulled from the 13 spades in a standard deck of cards.
Given that,
There are 13 spades in the standard deck of cards.
We need to pull 4 spades out of 13 spades.
First spade:
1. The first spade has to be from the standard deck of cards.
2. The first spade can be chosen in 13 ways.
Second spade:
1. The first spade is chosen from the deck of cards in 13 ways.
2. As the first spade is already being chosen, the second spade can be chosen in 12 ways.
Third spade:
1. The first spade is chosen in 13 ways and the second spade in 12 ways.
2. As the first spade and second spade are already being chosen, the third spade can be chosen in 11 ways.
Fourth spade:
1. The first spade is chosen in 13 ways, the second spade in 12 ways and the third spade in 11 ways.
2. As the first spade, second spade, and third spade are already being chosen, the fourth spade can be chosen in 10 ways.
As per the definition of combinations,
The total number of combinations = ways to choose a first spade $\times$ ways to choose a
second spade $\times$ ways to choose the third spade $\times$ ways to choose the fourth spade.
Which is given by,
$\Rightarrow 13\times 12\times 11\times 10$
$\Rightarrow 156\times 110$
$\Rightarrow 17160$

Therefore, 17160 combinations of 4 spades can be pulled from the 13 spades in a standard deck of cards.

Note: We need to know the basic concepts and definitions of combinations to solve this problem. Combinations usually are the number of ways the objects can be selected without replacement. Common mistakes like double counting, confusion of values should be avoided to get precise results.