Count the number of ways in which four cards, each of a different face value, can be chosen.

Sure, I can help with that.

The problem is asking for the number of ways to choose 4 cards, each of a different face value.

A standard deck of cards has 52 cards: 4 suits (hearts, diamonds, clubs, spades) each with 13 face values (Ace through 10, and the face cards Jack, Queen, King).

Since we want 4 cards each of a different face value, we first choose the 4 face values and then choose the suit for each of those face values.

Step 1: Choose the face values. There are 13 face values and we want to choose 4 of them. The number of ways to do this is given by the combination formula "n choose k", which is n! / [k!(n-k)!], where n is the total number of options, k is the number of options to choose, and "!" denotes factorial. So, there are 13! / [4!(13-4)!] = 715 ways to choose the face values.

Step 2: Choose the suit for each face value. For each face value, there are 4 suits to choose from. Since we have 4 face values, the number of ways to choose the suits is 4^4 = 256.

Therefore, the total number of ways to choose 4 cards each of a different face value is 715 * 256 = 183,040.