SOLUTION: How many ways are there to choose five cards from 52 so that you have two cards with the same number?

Did you mean EXACTLY two cards with the same number or
did you mean AT LEAST two cards with the same number?
We cannot tell.

-----------------------------------------------------
If you meant EXACTLY two cards with the same number:

Choose the single number to have exactly two cards of in 13 ways.
Choose the suits for those two cards in 4C2 = 6 ways
Choose three of the remaining numbers for the other 3 cards 
in 12C3=220 ways
Choose the suit for the smallest numbered card of those three in 4 ways.
Choose the suit for the middle numbered card of those three in 4 ways.
Choose the suit for the largest numbered card of those three in 4 ways.

That's 13∙6∙220∙4∙4∙4 = 1098240 ways.
------------------------------------
If you meant AT LEAST two cards with the same number:

We first calculate the number of ways for the complement event,
the number of ways where there are no two cards with the same number:

Choose 5 different numbers from the 13 in 13C5 = 1287 ways.
Choose the suit for the smallest numbered card in 4 ways.
Choose the suit for the next to smallest numbered card in 4 ways.
Choose the suit for the middle numbered card in 4 ways.
Choose the suit for the next to largest numbered card in 4 ways.
Choose the suit for the largest numbered card in 4 ways.

That's 1287∙4∙4∙4∙4∙4 = 1317888 ways for the complement event.

We can choose any 5 cards in 52C5 = 2598960 ways.  We subtract the
number of ways the complement event can occur:

2598960 - 1317888 = 1281072 ways.

Edwin