Question

There are 12 face cards in a standard deck of 52 cards. How many ways can you arrange a standard deck of 52 cards such that the first card is a face card?

Answer 1

`12(1!)(51!)=1.86xx10^67`

Explanation:

There are a few ways to approach this - I'm going to use the one that makes sense to me (and hopefully to you!)

We start with knowing there are 52 cards in a deck of cards. We are asked for the number of ways we can arrange the cards such that a face card is the first. (Face cards are the King, Queen, and Jack of each of the four suits: Spades, Hearts, Clubs, Diamonds).

Let's start with the King of Spades as the first card. We can now arrange the remaining 51 cards any way we want. When we arrange cards any way we want, we use the Factorial calculation. So that's 1 card arranged any way we want multiplied by 51 cards arranged any way we want:

`1!(51!)`

We can do this calculation 12 times - once for each of the 12 Face Cards in the front of the deck, so that puts us at:

`12(1!)(51!)`

`51!` is a big number so we'll end up with an approximation for the entire calculation of `1.86xx10^67` different ways. Compare this to the total number of ways to arrange a full deck of cards at `8.07xx10^67`

Which is an amazingly huge number. As an exercise as to just how big this number is, read the link below:

http://czep.net/weblog/52cards.html