Question

Hanafudais a Japanese game that uses a deck of cards made up of 12 suits, with each suit having four cards. How many 7-card hands can be formed so that 3 are from one suit and 4 are from another?

Short answer

The number of ways the 7-card hands can be drawn is958003200.

Step-by-step solution

Step 1. Given Information.

Given Hanafuda is a Japanese game that uses a deck of cards made up of 12 suits, with each suit having four cards. The number of 7-card hands that can be formed so that 3 are from one suit and 4 are from another is to be determined.

Step 2. Calculation.

A permutation is when n objects are available and r are to be picked and arranged in a certain order and the number of permutations is given by $P\left(n,r\right)=\frac{n!}{\left(n-r\right)!}$

A combination is when n objects are available and r are to be picked without arrangement and the number of combinations is given by $C\left(n,r\right)=\frac{n!}{\left(n-r\right)!r!}$

Here, first two of the 12 suits is to be arranged as one suit gives three cards and another suit gives four cards which can be done in $P\left(12,2\right)$ ways.

Three cards of the four cards in a suit can be selected in $C\left(4,3\right)$ ways.

Four cards in the second suit can be selected in $C\left(4,4\right)$ ways.

Hence the total number of ways is given by:

$\begin{array}{l}P\left(12,2\right)\cdot C\left(4,3\right)\cdot C\left(4,4\right)=\frac{12!}{2!}\cdot \frac{4!}{\left(4-3\right)!3!}\cdot \frac{4!}{\left(4-4\right)!4!}\\ P\left(12,2\right)\cdot C\left(4,3\right)\cdot C\left(4,4\right)=\frac{12!}{2!}\cdot \frac{4!}{1!3!}\cdot \frac{4!}{0!4!}\\ P\left(12,2\right)\cdot C\left(4,3\right)\cdot C\left(4,4\right)=\frac{12\cdot 11\cdot 10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2!}{2!}\cdot \frac{4\cdot 3!}{1\cdot 3!}\cdot \frac{4!}{1\cdot 4!}\\ P\left(12,2\right)\cdot C\left(4,3\right)\cdot C\left(4,4\right)=12\cdot 11\cdot 10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 4\\ P\left(12,2\right)\cdot C\left(4,3\right)\cdot C\left(4,4\right)=958003200\end{array}$

Step 3. Conclusion.

Hence, the number of ways the 7-card hands can be drawn is 958003200.