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.

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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.
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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