Vanilla ice cream

Alice stopped at an ice cream stand. Today they had pineapple, chocolate, yogurt, punch, vanilla, and apple ice cream on offer. Alice wants to buy two scoops of different ice cream. How many fewer options does Alice have when choosing ice cream if she knows that she will definitely not choose punch ice cream? The order of the scoops in the cone does not matter to her.

Correct answer:

x =  5

Step-by-step explanation:

$$\begin{gathered} C_2(6)=\left(\genfrac{}{}{0ex}{}{6}{2}\right)=\frac{6!}{2!(6-2)!}=\frac{6\cdot 5}{2\cdot 1}=15 \\ {C}_2(5)=\left(\genfrac{}{}{0ex}{}{5}{2}\right)=\frac{5!}{2!(5-2)!}=\frac{5\cdot 4}{2\cdot 1}=10 \\ {x}_1=\left(\genfrac{}{}{0ex}{}{2}{6}\right)=15 \\ {x}_2=\left(\genfrac{}{}{0ex}{}{2}{6-1}\right)=10 \\ x=x_1-x_2=15-10=5 \end{gathered}$$

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