SOLUTION: If you were to draw a 5-card hand from a standard deck of 52 cards, what is the probability that it will have 3 face cards (from the 4 jacks, 4 queens, and 4 kings)?

It is not clear whether the words "it will have 3 face cards" means 
"EXACTLY 3 and no more" or whether it means "AT LEAST 3".

I will considering that it means getting EXACTLY 3 face cards and no more.

The denominator of the desired probability in either case is
the number of ways to choose 5 cards from the 52: 

52 Choose 5 = 52C5 = 

After cancelling and multiplying or multiplying and dividing, we get 2598960

The numerator of the desired probability in this case is
the number of ways to choose 3 cards from the 12 face cards and
2 cards from the other 40 non-face cards 

12 Choose 3 = 12C3 = 

After cancelling and multiplying or multiplying and dividing, we get 220

For each of those ways to get exactly 3 face cards, there are 40C2 ways
to choose 2 non-face cards to go with the 3 face cards.

40 Choose 2 = 40C2 = 

After cancelling and multiplying or multiplying and dividing, we get 

So the number of possible hands with EXACTLY 3 face cards and EXACTLY 2 non-face cards is 220*780 = 171600

So the probability of getting EXACTLY 3 face cards and EXACTLY 2 non-face cards is

 which reduces to  or about 6.6% of the time.

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If your problem means "AT LEAST 3 face card", then we have to find the
probability of getting EXACTLY 4 face cards and EXACTLY 1 non-face card.

AND

the probability of getting ALL 5 face cards.

and add those two probabilities to the 

These will be found similar to the other case.

Edwin