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.