A person has 8 friends, of whom 5 will be invited to a party.

# A person has 8 friends, of whom 5 will be invited to a party.

J
240 points

A person has 8 friends, of whom 5 will be invited to a party.
(a) How many choices are there if 2 of the friends are feuding and will not attend together?
(b) How many choices if 2 of the friends will only attend together?

A person
jeffp

8.7k points

part (a) The total number of choices will be choosing 5 from the 8 order not mattering minus those groups who include the feuding friends,

$$\begin{pmatrix} 8 \\ 5 \end{pmatrix} - \begin{pmatrix} 2 \\ 2 \end{pmatrix} \begin{pmatrix} 6 \\ 3 \end{pmatrix} = 36$$

part (b) If the two friends are not included in the group than we can get $\begin{pmatrix} 6 \\ 5 \end{pmatrix}$ friends. If the two friends are included then we can get $\begin{pmatrix} 2 \\ 2 \end{pmatrix} \begin{pmatrix} 6 \\ 3 \end{pmatrix}$ friends. These are mutually exclusive events for a total of,

$$\begin{pmatrix} 6 \\ 5 \end{pmatrix} + \begin{pmatrix} 2 \\ 2 \end{pmatrix} \begin{pmatrix} 6 \\ 3 \end{pmatrix} = 26$$

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