Mumbai

A job placement agency in Mumbai had to send ten students to five companies, two to each. Two of the companies are in Mumbai, and others are outside. Two students prefer to work in Mumbai, while three prefer to work outside. How many ways can assignments be made if preferences are satisfied?

Correct answer:

n =  682

Step-by-step explanation:

$$\begin{gathered} C_2(7)=\left(\genfrac{}{}{0ex}{}{7}{2}\right)=\frac{7!}{2!(7-2)!}=\frac{7\cdot 6}{2\cdot 1}=21 \\ {C}_2(5)=\left(\genfrac{}{}{0ex}{}{5}{2}\right)=\frac{5!}{2!(5-2)!}=\frac{5\cdot 4}{2\cdot 1}=10 \\ {C}_2(2)=\left(\genfrac{}{}{0ex}{}{2}{2}\right)=\frac{2!}{2!(2-2)!}=\frac{1}{1}=1 \\ a=\left(\genfrac{}{}{0ex}{}{2+5}{2}\right)+\left(\genfrac{}{}{0ex}{}{5}{2}\right)=\left(\genfrac{}{}{0ex}{}{2+5}{2}\right)+10=31 \\ {C}_2(6)=\left(\genfrac{}{}{0ex}{}{6}{2}\right)=\frac{6!}{2!(6-2)!}=\frac{6\cdot 5}{2\cdot 1}=15 \\ {C}_2(4)=\left(\genfrac{}{}{0ex}{}{4}{2}\right)=\frac{4!}{2!(4-2)!}=\frac{4\cdot 3}{2\cdot 1}=6 \\ b=\left(\genfrac{}{}{0ex}{}{6}{2}\right)+\left(\genfrac{}{}{0ex}{}{4}{2}\right)+\left(\genfrac{}{}{0ex}{}{2}{2}\right)=15+6+1=22 \\ n=a\cdot b=31\cdot 22=682 \end{gathered}$$

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