Honored students

Of the 25 students in the class, ten are honored. How many ways can we choose five students from them if there are to be exactly two honors between them?

Final Answer:

n =  20475

Step-by-step explanation:

$$\begin{gathered} C_3(15)=\left(\genfrac{}{}{0ex}{}{15}{3}\right)=\frac{15!}{3!(15-3)!}=\frac{15\cdot 14\cdot 13}{3\cdot 2\cdot 1}=455 \\ {C}_2(10)=\left(\genfrac{}{}{0ex}{}{10}{2}\right)=\frac{10!}{2!(10-2)!}=\frac{10\cdot 9}{2\cdot 1}=45 \\ {n}_1=\left(\genfrac{}{}{0ex}{}{10}{2}\right)=45 \\ {n}_2=\left(\genfrac{}{}{0ex}{}{25-10}{3}\right)=455 \\ n=n_1\cdot n_2=45\cdot 455=20475 \end{gathered}$$

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