Combinatorics: Party Attenders 2

A person has 8 friends and plans to invite 5 of them to a party. How many combinations are possible if 2 of those friends must either both attend together or both not attend?

Answer

We consider two cases:

  1. Neither of the 2 friends attends: Select all 5 guests from the remaining 6 friends.

    \[ \binom{6}{5} = 6 \]

  2. Both of the 2 friends attend: With these 2 friends included, select the remaining 3 guests from the other 6 friends.

    \[ \binom{6}{3} = 20 \]

Total combinations \(= 6 + 20 = 26\)

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