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There were 12 members on the commission. Five members were in favor, and seven members were against the proposal. In how many ways could it help the commission vote?

Correct answer:

n =  792

Step-by-step explanation:

$$\begin{gathered} C_5(12)=\left(\genfrac{}{}{0ex}{}{12}{5}\right)=\frac{12!}{5!(12-5)!}=\frac{12\cdot 11\cdot 10\cdot 9\cdot 8}{5\cdot 4\cdot 3\cdot 2\cdot 1}=792 \\ {C}_7(12)=\left(\genfrac{}{}{0ex}{}{12}{7}\right)=\frac{12!}{7!(12-7)!}=\frac{12\cdot 11\cdot 10\cdot 9\cdot 8}{5\cdot 4\cdot 3\cdot 2\cdot 1}=792 \\ t=12 \\ 5+7=12=t \\ n=\left(\genfrac{}{}{0ex}{}{t}{5}\right)=792 \\ \text{ Verifying Solution: } \\ {n}_2=\left(\genfrac{}{}{0ex}{}{t}{7}\right)=792 \end{gathered}$$

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