What is the number of ways of choosing 4 cards from a pack of 52 playing cards? In how many of these (i) four cards are of the same suit, (ii) four cards belong to four different suits, (iii) are face cards, (iv) two are red cards and two are black cards, (v) cards are of the same colour?

Step 1: Calculate total ways to choose 4 cards

To find the total number of ways to choose 4 cards from a standard deck of 52 cards, we use the combination formula $\binom{n}{k} = \frac{n!}{k!(n-k)!}$, where $n$ is the total number of items and $k$ is the number of items to choose. Here, $n=52$ and $k=4$.

Step 2: Calculate ways for four cards of the same suit

There are 4 suits (hearts, diamonds, clubs, spades), and each suit has 13 cards. To choose 4 cards of the same suit, we first choose 1 suit out of 4, and then choose 4 cards from the 13 cards of that chosen suit.

Step 3: Calculate ways for four cards from four different suits

To choose 4 cards such that each card belongs to a different suit, we select one card from each of the four suits. Since there are 13 cards in each suit, we have 13 choices for the card from the first suit, 13 for the second, and so on.

Step 4: Calculate ways for four face cards

A standard deck of 52 cards has 12 face cards (Jack, Queen, King for each of the 4 suits). To choose 4 face cards, we simply select 4 cards from these 12 face cards.

Step 5: Calculate ways for two red and two black cards

A deck of 52 cards has 26 red cards (hearts and diamonds) and 26 black cards (clubs and spades). To choose two red cards and two black cards, we choose 2 cards from the 26 red cards and 2 cards from the 26 black cards.

Step 6: Calculate ways for cards of the same colour

The cards can either all be red or all be black. There are 26 red cards and 26 black cards. So, we calculate the ways to choose 4 cards from 26 red cards and add it to the ways to choose 4 cards from 26 black cards.

Step 7: Compute the numerical values

Now we compute the numerical values for each of the combinations calculated in the previous steps. The combination formula is $\binom{n}{k} = \frac{n!}{k!(n-k)!}$.

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