"Of course, you have to answer three questions for me first," Alice told William with a playful glint in her eye.269Please respect copyright.ＰＥＮＡＮＡ5dQWUuQrmz
"What questions?" William asked, intrigued.269Please respect copyright.ＰＥＮＡＮＡlWhsS6I7eo
"At a gathering of 1179151902372306698608701103666282114628714499269Please respect copyright.ＰＥＮＡＮＡNpm5x1I3Hg
people, there are 786101268248204465739134069110854743085809666269Please respect copyright.ＰＥＮＡＮＡseL20VXHpS
people who all know each other and 393050634124102232869567034555427371542904833 people who know no one. People who know each other269Please respect copyright.ＰＥＮＡＮＡqrgvYtOxHo
hug, and people who do not know each other shake hands. How many269Please respect copyright.ＰＥＮＡＮＡZg0F9KeIbO
handshakes occur within the group?" Alice said with a playful glint in her eye.269Please respect copyright.ＰＥＮＡＮＡJF31jZTdlU
"I know the answer," said William confidently.
"How is that possible?" Alice exclaimed, surprised.
"In this gathering," William explained, "everyone falls into two groups: those who know everyone else and those who don't know anyone else. For each person, they can shake hands with everyone else except themselves, which means each person can have handshakes with (1,179,151,902,372,306,698,608,701,103,666,282,114,628,714,499 - 1) individuals."
"Let's calculate the total number of hugs first," he continued. "For those who know each other, each person can hug every other person except themselves. So, the total number of hugs is calculated by the formula: hugs = (total people × (total people - 1)) / 2, where the total number of people is 786,101,268,248,204,465,739,134,069,110,854,743,085,809,666."
"Next, for those who don't know anyone else," Alice added, "they can shake hands with everyone except themselves, as they know no one else in the group. Using the formula: handshakes = (total people who know no one × (total people who know no one - 1)) / 2, where the total number of people who know no one is 393,050,634,124,102,232,869,567,034,555,427,371,542,904,833."
"In the end," William concluded, "we add the total number of hugs and handshakes to find the overall number of interactions within the gathering."
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