To solve this problem, we need to determine the number of ways of choosing 4 cards from a standard deck of 52 playing cards, and then break it down into different cases.
Total number of ways to choose 4 cards from 52 cards:
This is a basic combination problem, where the number of ways to choose 4 cards from 52 is given by the combination formula:
C(n, r) = n! / [r!(n - r)!]
Here, n = 52 (total cards) and r = 4 (cards to choose). So, the total number of ways to choose 4 cards is:
C(52, 4) = 52! / [4!(52 - 4)!] = (52 × 51 × 50 × 49) / (4 × 3 × 2 × 1) = 270,725.
So, there are 270,725 ways to choose 4 cards from a deck of 52 cards.
Now, let’s solve for the other cases:
(i) Four cards are of the same suit:
To choose 4 cards from the same suit, we first choose one suit (there are 4 suits: hearts, diamonds, clubs, and spades). For each suit, we need to choose 4 cards out of the 13 available cards in that suit.
The number of ways to choose 4 cards from a single suit is:
C(13, 4) = (13 × 12 × 11 × 10) / (4 × 3 × 2 × 1) = 715.
Since there are 4 suits, the total number of ways to choose 4 cards of the same suit is:
4 × 715 = 2,860.
(ii) Four cards belong to four different suits:
If we need to select 4 cards, each from a different suit, we first choose 4 suits (one for each card). There is only 1 way to assign a suit to each card (since each suit must be represented). Then, we select one card from each suit (13 options for each suit).
So, the number of ways to choose 4 cards from different suits is:
C(13, 1) × C(13, 1) × C(13, 1) × C(13, 1) = 13 × 13 × 13 × 13 = 13^4 = 28,561.
(iii) The cards are face cards:
In a deck of 52 playing cards, there are 12 face cards (Jack, Queen, and King in each of the 4 suits). We need to choose 4 face cards from these 12.
The number of ways to choose 4 face cards from the 12 available is:
C(12, 4) = (12 × 11 × 10 × 9) / (4 × 3 × 2 × 1) = 495.
(iv) Cards are of the same colour:
There are two colors in a deck of cards: red (hearts and diamonds) and black (clubs and spades). Each color has 26 cards (13 hearts + 13 diamonds for red, and 13 clubs + 13 spades for black).
We need to choose 4 cards all of the same color. This can happen in two ways: either all 4 cards are red, or all 4 cards are black.
For red cards, the number of ways to choose 4 from the 26 red cards is:
C(26, 4) = (26 × 25 × 24 × 23) / (4 × 3 × 2 × 1) = 14,950.
For black cards, the number of ways to choose 4 from the 26 black cards is:
C(26, 4) = 14,950.
So, the total number of ways to choose 4 cards of the same color (either red or black) is:
14,950 (red) + 14,950 (black) = 29,900.
(v) Two red cards and two black cards:
There are 26 red cards and 26 black cards in the deck. We need to choose 2 red cards and 2 black cards.
The number of ways to choose 2 red cards from the 26 red cards is:
C(26, 2) = (26 × 25) / (2 × 1) = 325.
The number of ways to choose 2 black cards from the 26 black cards is:
C(26, 2) = 325.
So, the total number of ways to choose 2 red and 2 black cards is:
325 × 325 = 105,625.
Summary of answers:
Total number of ways to choose 4 cards from 52 cards: 270,725.
Number of ways to choose 4 cards of the same suit: 2,860.
Number of ways to choose 4 cards from 4 different suits: 28,561.
Number of ways to choose 4 face cards: 495.
Number of ways to choose 4 cards of the same color: 29,900.
Number of ways to choose 2 red cards and 2 black cards: 105,625.
