Problem Solving

In the document downloadable from here:

https://www.beatthegmat.com/difficult-gm ... s-t78.html

this question appears:

20. In how many ways can one choose 6 cards from a normal deck of cards so as to have all suits present?

a. (13^4) x 48 x 47
b. (13^4) x 27 x 47
c. 48C6
d. 13^4
e. (13^4) x 48C6


I know this isn't an official GMAT question, but would you read that as implying that order matters, or that order does not matter?

I read it as:

How many ways can you draw 6 cards from 52 and have at least one card of each suit?

But it could also be read as:

How many 6-card hands from a standard deck contain at lease one card of each suit? The answer given in the text was [spoiler]13^4 *48*47[/spoiler].


I read it the first way, and tried to solve it like this:

There are two possible types of suit arrangement:

- 3 of one, and 1 of each of the three others
- 2 of one, 2 of another, and one each of the other two.

The first arrangement can occur in (4C1) four ways, the second arrangement can occur in (4C2) six ways.

So total possibilities =

4 * (13C3 * 13^3) + 6 * (13C2 * 13C2 * 13^2)

= 4*(13 * 2 * 11 * 13^3) + 6 (13 * 6 * 13 * 6 * 13^2)

= 88*(13^4) + 216(13^4)

= 304 * (13^4)

Does that look right, given my assumption about the meaning of the question?