# [SOLVED] Card Combinations

• Feb 19th 2009, 01:50 PM
[SOLVED] Card Combinations
If a bridge hand consists of 13 cards, how many bridge hands include 5 cards of one suit, 6 cards of a second suit, and 2 cards of another suit?

Since we don't care what the cards are and just what their suit is would the answer be 4P3 since the order of the suits matters since there are a different number of each cards of a particular suit?
• Feb 19th 2009, 07:38 PM
Soroban

Quote:

If a bridge hand consists of 13 cards, how many bridge hands include:
5 cards of one suit, 6 cards of a second suit, and 2 cards of another suit?

First, select 3 of the 4 suits and place them in an order.

[The first suit is the 5-card set, the second is the 6-card set, the third the 2-card set.]
. . There are: . $_4P_3$ ways.

Then we will choose:

. . $\begin{array}{cc}\text{5 of the 1st suit:} &_{13}C_5\text{ ways.} \\ \\[-4mm]
\text{6 of the 2nd suit:} & _{13}C_6 \text{ ways.} \\ \\[-4mm]
\text{2 of the 3rd suit:} & _{13}C _2 \text{ ways.}
\end{array}$

Therefore, there are: . $\left(_4P_3\right)\left(_{13}C_5\right)\left(_{13} C_6\right)\left(_{13}C_2\right)$

. . . . . . . . . . . . . . $= \;(24)(1,\!287)(1,\!716)(78)$

. . . . . . . . . . . . . . $= \;4,\!134,\!297,\!024\text{ hands.}$