A face card is a card that is a King, a Queen or a Jack. In a standard deck, there is a total of 12 face cards.
Suppose that the first 7 cards we take are faces and the remaining two are not.
The probability of taking a face on the first withdrawal, is 12/52, since there are 12 faces and there is a total of 52 cards.
On the second withdrawal, there are 11 faces remaining, and a total of 51 cards remain. Therefore, the probability of the second card also being a face is 11/51.
On the third withdrawal, the probability of taking a face will be 10/50, and so on:
On the fourth withdrawal, the probability will be 9/49.
On the fifth, 8/48.
On the sixth, 7/47.
On the seventh, 6/46.
The next two withdrawals must not be faces. On the eight withdrawal, a total of 45 cards remain, 5 of them are faces. So, the probability of not taking a face, is 40/45.
On the nineth withdrawal, the probability of not taking a face is 39/44.
The combined probability of all this events happening one after another is the product of the individual probabilities. Therefore, the probability of the hand containing exactly 7 face cards, is:
[tex]\frac{12}{52}\cdot\frac{11}{51}\cdot\frac{10}{50}\cdot\frac{9}{49}\cdot\frac{8}{48}\cdot\frac{7}{47}\cdot\frac{6}{46}\cdot\frac{40}{45}\cdot\frac{39}{44}[/tex]Write each factor as a product of powers of prime numbers:
[tex]=\frac{2^2\cdot3}{2^2\cdot13}\cdot\frac{11}{3\cdot17}\cdot\frac{2\cdot5}{2\cdot5^2}\cdot\frac{3^2}{7^2}\cdot\frac{2^3}{2^4\cdot3}\cdot\frac{7}{47}\cdot\frac{2\cdot3}{2\cdot23}\cdot\frac{2^3\cdot5}{3^2\cdot5}\cdot\frac{3\cdot13}{2^2\cdot11}[/tex]Use the laws of exponents to reduce this expression:
[tex]\begin{gathered} =\frac{2^{10}\cdot3^5\cdot5^2\cdot7\cdot11\cdot13}{2^{10}\cdot3^4\cdot5^3\cdot7^2\cdot11\cdot13\cdot17\cdot23\cdot47} \\ =\frac{3}{5\cdot7\cdot17\cdot23\cdot47} \\ =\frac{3}{643195} \end{gathered}[/tex]Therefore, the probability is:
[tex]\frac{3}{643195}[/tex]
