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09 Jun 2026, 07:12
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I was assuming there is no way to seat 7 people in 5 chairs (question says seating x people in y chairs) why are we assuming it is fine to not seat 2 people?
09 Jun 2026, 15:46
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Nidhibatra
If x = 7 and y = 5, we choose which 5 of the 7 people get seats and arrange those 5 in the 5 chairs. That is why this case is still counted.
10 Jun 2026, 09:34
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Hi Nidhibatra,
Your instinct makes sense: if the task were "every single person must get a chair," then 7 people with only 5 chairs would be impossible, and the x=7, y=5 case would just collapse. But that stricter reading isn't what the question is asking.
Look closely at the phrasing in the prompt the whole thread is working from: "How many ways can the x people be seated in the y chairs (assuming that each chair can seat exactly one person)." The only hard rule given is one person per chair. There is no rule that says every person must be seated.
So when people outnumber chairs (7 people, 5 chairs), we simply fill the 5 chairs - and the 2 people who don't get a chair just stand. That's not a forbidden outcome; it's a normal one. We are counting the ways to fill the available chairs, which is the same as choosing which 5 of the 7 people sit and then arranging them.
Therefore that's exactly why Dmitry's count works out to 7·6·5·4·3 either way:
- 5 people, 7 chairs: each person picks an empty chair - 7·6·5·4·3.
- 7 people, 5 chairs: each chair gets one of the people - 7·6·5·4·3.
A tiny version makes it feel natural. Say 3 people, 2 chairs. We don't throw our hands up because someone is left standing - we just count: chair 1 can hold any of 3 people, chair 2 any of the remaining 2, giving 6 ways. The third person stands, and that's fine.
So "seating people in chairs" here means assigning people to the chairs that exist - leftover people standing is allowed, which is why x=7, y=5 is a perfectly valid scenario and Statement (1) still pins down a single answer.
Answer: A
Your instinct makes sense: if the task were "every single person must get a chair," then 7 people with only 5 chairs would be impossible, and the x=7, y=5 case would just collapse. But that stricter reading isn't what the question is asking.
Look closely at the phrasing in the prompt the whole thread is working from: "How many ways can the x people be seated in the y chairs (assuming that each chair can seat exactly one person)." The only hard rule given is one person per chair. There is no rule that says every person must be seated.
So when people outnumber chairs (7 people, 5 chairs), we simply fill the 5 chairs - and the 2 people who don't get a chair just stand. That's not a forbidden outcome; it's a normal one. We are counting the ways to fill the available chairs, which is the same as choosing which 5 of the 7 people sit and then arranging them.
Therefore that's exactly why Dmitry's count works out to 7·6·5·4·3 either way:
- 5 people, 7 chairs: each person picks an empty chair - 7·6·5·4·3.
- 7 people, 5 chairs: each chair gets one of the people - 7·6·5·4·3.
A tiny version makes it feel natural. Say 3 people, 2 chairs. We don't throw our hands up because someone is left standing - we just count: chair 1 can hold any of 3 people, chair 2 any of the remaining 2, giving 6 ways. The third person stands, and that's fine.
So "seating people in chairs" here means assigning people to the chairs that exist - leftover people standing is allowed, which is why x=7, y=5 is a perfectly valid scenario and Statement (1) still pins down a single answer.
Answer: A
Nidhibatra
