The face cards are 4 Jacks, 4 Queens and 4 Kings.
That's 12 face cards.

We can successfully choose 4 face cards any of 12C4=495 ways.
There are 52-12=40 non-face cards.
We can choose the other 3 non-face cards in any of 40C3=9880 ways.

That's (495)(9880)=4890600 ways

We can possibly choose any 7 cards in 52C7=133784560 ways.

That's 4980600 successful ways out of 133784560 possible ways.

The probability is 4890600/133784560 which reduces to 9405/257278.
That's a probability of about 3.7%.

Edwin