Bridge cards

How many bridge hands are possible containing 4 spades,6 diamonds, 1 club, and 2 hearts?

Correct result:

n =  1244117160

Solution:

$$\begin{gathered} C_4(13)=\left({\displaystyle \genfrac{}{}{0px}{}{13}{4}}\right)={\displaystyle \frac{13!}{4!(13-4)!}}={\displaystyle \frac{13\cdot 12\cdot 11\cdot 10}{4\cdot 3\cdot 2\cdot 1}}=715 \\ {C}_6(13)=\left({\displaystyle \genfrac{}{}{0px}{}{13}{6}}\right)={\displaystyle \frac{13!}{6!(13-6)!}}={\displaystyle \frac{13\cdot 12\cdot 11\cdot 10\cdot 9\cdot 8}{6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}}=1716 \\ {C}_1(13)=\left({\displaystyle \genfrac{}{}{0px}{}{13}{1}}\right)={\displaystyle \frac{13!}{1!(13-1)!}}={\displaystyle \frac{13}{1}}=13 \\ {C}_2(13)=\left({\displaystyle \genfrac{}{}{0px}{}{13}{2}}\right)={\displaystyle \frac{13!}{2!(13-2)!}}={\displaystyle \frac{13\cdot 12}{2\cdot 1}}=78 \\ s=4+6+1+2=13 \\ n=\left(\genfrac{}{}{0px}{}{s}{4}\right)\cdot \left(\genfrac{}{}{0px}{}{s}{6}\right)\cdot \left(\genfrac{}{}{0px}{}{s}{1}\right)\cdot \left(\genfrac{}{}{0px}{}{s}{2}\right)=715\cdot 1716\cdot 13\cdot 78=1244117160=1.244\cdot 10^9 \end{gathered}$$

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Math student

3{2*-2}-4{2*-1}<12

2 years ago  Like

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