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?

Correct answer:

n =  12870

Step-by-step explanation:

$$\begin{gathered} C_3(13)=\left(\genfrac{}{}{0ex}{}{13}{3}\right)=\frac{13!}{3!(13-3)!}=\frac{13\cdot 12\cdot 11}{3\cdot 2\cdot 1}=286 \\ {C}_2(10)=\left(\genfrac{}{}{0ex}{}{10}{2}\right)=\frac{10!}{2!(10-2)!}=\frac{10\cdot 9}{2\cdot 1}=45 \\ {n}_1=\left(\genfrac{}{}{0ex}{}{10}{2}\right)=45 \\ {n}_2=\left(\genfrac{}{}{0ex}{}{25-10-2}{3}\right)=286 \\ n=n_1\cdot n_2=45\cdot 286=12870 \end{gathered}$$

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