Answer to Question #235661 in Statistics and Probability for James

Question

Thirteen poker hands is to be taken from an ordinary deck of cards. How many ways it can contain two aces, three face cards, four cards greater than 2 but less than 5, and three cards greater than 4 but less than 10?

Answer 1

There are four aces in a deck from these, two aces can be selected in $4C_2$ ways

There are 12 face cards from which 3 cards can be chosen in $12C_3$ ways

A number greater than 2 and less than 5 are 3 and 4

There are $2\times4=8$ such cards from these 4 cards that can be chosen in $8C_4$ ways

Numbers greater than 4 and less than 10 are 5,6,7,8 and 9

There are $5\times4=20$ such cards from these 3 cards that can be chosen in $20C_3$ ways

The remaining 1 card can be selected from $52-12=40$ cards in $40C_1$ ways

So the total number of ways$=4C_2\times 12C_3\times8C_4\times20C_3\times40C_1=4213440000$ ways.